V —∆ =C —L—∆

Observe that V on the left is the subspace of elements of V — ∆ ¬xed by Id V — ι

and that C — L on the right is the subspace of elements of C — L — ∆ ¬xed by ιπ.

We only consider the case where char F = 2 and leave the case char F = 2 as an

exercise. Let V = Le ⊥ V 0 , V 0 = e⊥ , and let d ∈ L — ∆ be a generator of the

discriminant algebra ∆ such that (1 — ι)(d) = ’d. The map

φ: V — ∆ ’ C — L — ∆

given by φ ( e + v ) — s = 1 — — s + v — ds is such that φ —¦ ι = (π — ι) —¦ φ. Thus

the image of V in C — L — ∆ can be identi¬ed with L — 1 ⊥ C 0 — L — 1 and β is

the restriction of the twisted Hurwitz composition on C — L — ∆. This shows that

(??) implies (??).

(36.29) Corollary. If L is not a ¬eld, any twisted composition over L is similar

to a twisted Hurwitz composition “(C, L).

Proof : The claim for compositions over F — F — F follows from Proposition (??) (a

result used in the proof of Theorem (??)). Let (V, L, Q, β) be a twisted composition

over L = F — ∆, ∆ a quadratic ¬eld extension of F . We have to check that

(V, L, Q, β) is similar to a composition with an element e such that β(e) = e.

By decomposing V = (V0 , V1 ), Q = (Q0 , Q1 ) and β = (β0 , β1 ) according to the

decomposition F —∆, we see (compare with the proof of (??)) that β0 is a quadratic

map V1 ’ V0 such that Q0 β0 (v1 ) = nK/F Q1 (v1 ) and β1 is a ∆-linear map

ι

V1 — V0 ’ V1 such that Q1 β1 (v1 — v0 ) = ι Q1 (v1 ) Q0 (v0 ). Let bQ1 (x, y) be

the ∆-bilinear polar of Q1 and let bQ0 (x, y) be F -bilinear polar of Q0 as well as its

extension to V0 —∆ as a ∆-bilinear form. By an argument similar to the argument in

the proof of proposition (??), there exists a unique extension β0 : V1 —∆ι V1 ’ V0 —∆

§36. TWISTED COMPOSITIONS 503

of β0 as a ∆-hermitian map. Property (??) of the de¬nition of a twisted composition

implies that

(36.30) bQ0 v0 , β0 (v1 ) = bQ1 v1 , β1 (v1 — v0 ) for v0 ∈ V0 , v1 ∈ V1 .

For v = (v0 , 0), v0 = 0 in V0 we have bQ v, β(v) = 0, so that the claim follows

from Theorem (??).

(36.31) Remark. In the next section we give examples of twisted compositions

of dimension 8 over a ¬eld L which are not induced by Cayley algebras.

(36.32) Remark. Let “ = (V, L, Q, β) be a twisted composition. Since L — L

L—L—∆ (where the ¬rst projection is multiplication), “—L is similar to a Hurwitz

twisted composition “(C, L — L) for some unique Hurwitz algebra C(“) over L.

The algebra C(“) over L is in fact extended from a Hurwitz algebra C over F .

In dimension 8, the algebra C is determined by a cohomological invariant f3 (See

Proposition (??)).

36.D. Twisted compositions of type A2 . We ¬nish this chapter by showing

how to associate twisted compositions to symmetric compositions arising from cen-

tral simple algebras of degree 3 or cubic ´tale algebras. We assume that char F = 3

e

and ¬rst suppose that F contains a primitive cube root of unity ω. Let » ∈ F — and

√ √

let L = F [X]/(X 3 ’») = F ( 3 ») where 3 » is the class of X modulo (X 3 ’»). Since

√

char F = 3 and µ3 ‚ F — , F ( 3 ») is a cyclic cubic F -algebra and u ’ ωu de¬nes a

generator of Gal(L/F ). Let A be a central simple F -algebra of degree three or a

cubic ´tale algebra and let (A0 , n, ) be the corresponding composition algebra as

e

de¬ned in Proposition (??). As in Example (??) we de¬ne a cyclic composition on

A0 — L by

x — y = [(1 — ρ)(x)] [(1 — ρ2 )(y)]

√

and Q(x) = (n — 1)(x). Let v = 3 ». Taking (1, v, v ’1 ) as a basis of L over F , we

can write any element of A0 — L as a sum x = a + bv + cv ’1 with a, b, c ∈ A0 , and

we have, (using that µω 2 + ω(1 ’ µ) = 0 and µω + ω 2 (1 ’ µ) = ’1),

β(x) = x — x = (a + bω 2 v + cωv ’1 ) —1(a + bωv + cω 2 v ’1 )

= a2 ’ 3 TA (a2 ) ’ bc + 3 TA (bc)

1 1

+ v[»’1 c2 ’ 3 TA (c2 ) ’ ab + 1 TA (ab)]

1

3

+ v ’1 [» b2 ’ 1 TA (b2 ) ’ ca + 1 TA (ca)]

3 3

= a a ’ (bc)0 + v»’1 [c c ’ (ba)0 ] + v ’1 »[b b ’ (ca)0 ]

where x0 = x ’ 1 TA (x) for x ∈ A. The form Q is given by

3

Q(a + bv + cv ’1 ) = ’ 3 SA—L (a + bv + cv ’1 )

1

1

= ’ 3 [SA (a) + SA (c)»’1 v + SA (b2 )»v ’1 ]

1

+ 3 [TA (bc) + TA (ab)v + TA (ca)v ’1 ]

504 VIII. COMPOSITION AND TRIALITY

and the norm N by

N (a + bv + cv ’1 ) = bQ a + bv + cv ’1 , β(a + bv + cv ’1 )

= bn (a, a a) + »bn (b, b b) + »’1 bn (c, c c)

+ 1 [bSA a, (bc)0 + bSA b, (ca)0 + bSA c, (ab)0 ]

3

= NA (a) + »NA (b) + »’1 NA (c) ’ TA (abc)

since bn (a, a a) = NA (a) and BSA a, (bc)0 = ’T a, (bc)0 = ’TA (abc).

Assume now that F does not necessarily contain a primitive cube root of unity

ω. Replacing F by F (ω) = F [X]/(X 2 + X + 1), we may de¬ne — on A0 — F (ω) — L.

However, since ω does not explicitly appear in the above formulas for β and Q

restricted to A0 —L, we obtain for any algebra A of degree 3 over F of characteristic 3

and for any » ∈ F — a twisted composition “(A, ») = (A0 — L, L, Q, β) over L =

√

F ( 3 »). A twisted composition “ similar to a composition “(A, ») for A associative

central simple and » ∈ F — is said to be a composition of type 1A2 .

Any pair (φ, ψ) ∈ AutF (A) — AutF (L) induces an automorphism of “(A, L).

Thus we have a morphism of group schemes PGL3 —µ3 ’ Spin8 S3 and:

(36.33) Proposition. Twisted compositions “(A, ») of type 1A2 are classi¬ed by

the image of H 1 (F, PGL3 —µ3 ) in H 1 (F, Spin8 S3 ).

(36.34) Remark. If F contains a primitive cube root of unity, µ3 = A3 and

the image under the morphism PGL3 —µ3 ’ Spin8 S3 of the group PGL3 =

Spin8 A3 is contained in Spin8 .

Let now (B, „ ) be central simple of degree 3 over a quadratic ´tale F -algebra K,

e

—

with a unitary involution „ . For ν ∈ K we have a twisted K-composition “(B, ν)

√

over K( 3 ν) which we would like (under certain conditions) to descent to a twisted

F -composition.

(36.35) Proposition. If NK (ν) = 1, then:

√

(1) There is an ι-semilinear automorphism ι of K( 3 ν) of order 2 which maps ν

to ν ’1 ; its set of ¬xed elements is a cubic ´tale F -algebra L with disc(L) K —F (ω)

e

(where ω is a primitive cube root of unity).

(2) There is an ι-semilinear automorphism of order 2 of the twisted K-composi-

tion “(B, „ ) such that its set of ¬xed elements is a twisted F -composition “(B, „, ν)

Sym(B, „ )0 • B 0 ; under this isomorphism we have, for

over L with “(B, „, ν) √ √

z = (x, y) ∈ Sym(B, „ )0 • B 0 and v = 3 ν ∈ K( 3 ν) = L — K,

SB (x) + TL—K/L SB (y)»v ’1 + TB (xy)v + TB y„ (y) ,

1

Q(z) = 3

β(z) = x2 ’ 3 TB (x2 ) ’ y„ (y) + 3 TB y„ (y)

1 1

+ ν[„ (y)2 ’ 3 TB „ (y)2 ] ’ xy + 1 TB (xy),

1

3

N (z) = NB (x) + νNB (y) + ν ’1 NB „ (y) ’ TB xy„ (y) .

Proof : (??) This is Proposition (??). √

(??) We have “(B, ν) = B 0 —K K( 3 ν) and take as our „ -semilinear automor-

√

phism the map „ = „ — „ . We write B 0 —K K( 3 ν) = B 0 • B 0 v • B 0 v ’1 . The

isomorphism Sym(B, „ )0 •B 0 “(B, „, ν) is then given by (x, y) ’ x+yv+„ (y)v ’1

and it is easy to check that its image lies in the descended object “(B, „, ν). The

formulas for Q, β, and N follow from the corresponding formulas for type 1A2 .

§36. TWISTED COMPOSITIONS 505

(36.36) Example. If K = F (ω), ω a primitive cubic root of 1, then the L given

by Proposition (??) is cyclic and the twisted composition “(B, „, ν) is the twisted

composition associated to the cyclic composition Sym(B, „ )0 — L.

A twisted composition isomorphic to a composition “(B, „, ν) is said to be of

type 2A2 . We have a homomorphism

GU3 (K) — (µ3 )γ ’ Spin8 S3

where γ is a cocycle de¬ning K and the analogue of Proposition (??) is:

(36.37) Proposition. Twisted compositions “(B, „, ν) of type 2A2 are classi¬ed

by the image of H 1 F, PGU3 (K) — (µ3 )γ in H 1 (F, Spin8 S3 ).

36.E. The dimension 2 case. If (V, L, Q, β) is a twisted composition with

rankL V = 2, then V admits, in fact, more structure:

(36.38) Proposition. Let (V, L, Q, β) be a twisted composition with dim L V = 2.

There exists a quadratic ´tale F -algebra K which operates on V and a nonsingular

e

L — K-hermitian form h : V — V ’ L — K of rank 1 such that Q(x) = h(x, x),

NK — » where » can be chosen such that NL/F (») ∈ F —2 .

x ∈ V . Hence Q

Furthermore the algebra K is split if Q is isotropic.

Proof : For generic v ∈ V , we may assume that Q(v) = » ∈ L— , bQ v, β(v) =

a ∈ F — , and that {v, β(v)} are linearly independent over L (see also the following

Remark). Then v1 = v, v2 = »β(v) is an L-basis of V , and

Q(x1 v1 + x2 v2 ) = x2 + ax1 x2 + nL/F (»)x2 · »

1 2

Thus 4NL/F (») ’ a2 = detL Q is nonzero and the quadratic F -algebra

K = F [x]/ x2 + ax + NL/F (»)

is ´tale. Let ιK be the conjugation map of K. Let ξ = x+ x2 +ax+NL/F (») ∈ K.

e

We de¬ne a K-module structure on V by putting

ξv1 = v2 and ξv2 = ’av1 ’ NL/F (»)v2 .

Thus v = v1 is a basis element for the L — K-module V . We then de¬ne

h(·1 v, ·2 v) = ·1 »¯2

·

for ·1 , ·2 ∈ L — K, and ·1 = (1 — ιK )(·1 ). In particular we have » = h(v, v)

¯

for the chosen element v. The fact that Q(x) = h(x, x), x ∈ V , follows from the

formula Q(x1 v1 + x2 v2 ) = x2 + ax1 x2 + NL/F (»)x2 · ». The last claim, i.e., that

1 2

—2

NL/F (») ∈ F , follows by choosing v of the form v = β(u). If Q is isotropic, Q

is hyperbolic by Proposition (??) and Corollary (??), hence Q NK — 1 and the

claim follows from Springer™s theorem.

(36.39) Remark. If bQ v, β(v) = 0 for v = 0 or if {v, β(v)} is linearly dependent

over L, the twisted composition is induced from a Hurwitz algebra (see Theo-

rem (??)).

506 VIII. COMPOSITION AND TRIALITY

Exercises

1. Let C be a separable alternative algebra of degree 2 over F . Show that π : x ’ x

is the unique F -linear automorphism of C such that x + π(x) ∈ F · 1 for all

x ∈ C.

2. Let (C, N ), (C , N ) be Hurwitz algebras of dimension ¤ 4. Show that an

∼

isometry N ’ N which maps 1 to 1 is either an isomorphism or an anti-

’

isomorphism. Give an example where this is not the case for Cayley algebras.

3. A symmetric composition algebra with identity is 1-dimensional.

4. (Petersson [?]) Let K be quadratic ´tale with norm N = NK and conjugation

e

x ’ x. Composition algebras (K, ) are either K (as a Hurwitz algebra) or”up

to isomorphism”of the form

(a) x y = xy,

(b) x y = xy, or

(c) x y = uxy for some u ∈ K such that N (u) = 1.

Compositions of type (??) are symmetric.

5. The split Cayley algebra over F can be regarded as the set of all matrices (Zorn

matrices) ± β with ±, β ∈ F and a, b ∈ F 3 , with multiplication

a

b

±a γ c ±γ + a · d ±c + δa ’ (b § d)

=

bβ d δ γb + βd + (a § c) βδ + b · c

where a · d is the standard scalar product in F 3 and b § d the standard vector

product (cross product). The conjugation is given by

±a β ’a

π =

bβ ’b ±

and the norm by

±a

= ±β ’ a · b.

n

bβ

6. Let K be a quadratic ´tale F -algebra and let (V, h) be a ternary hermitian space

e

over K with trivial (hermitian) discriminant, i.e., there exists an isomorphism

∼

φ : §3 (V, h) ’ 1 . For any v, w ∈ V , let v — w ∈ V be determined by the

’

condition h(u, v — w) = φ(u § v § w).

(a) Show that the vector space C(K, V ) = K • V is a Cayley algebra under

the multiplication

(a, v) (b, w) = ab ’ h(v, w), aw + bv + v — w

and the norm n (a, v) = NK/F (a) + h(v, v).

(b) Conversely, if C is a Cayley algebra and K is a quadratic ´tale subalgebra,

e

⊥

then V = K admits the structure of a hermitian space over K and

C C(K, V ).

(c) AutF (C, K) = SU3 (K).

There exists a monomorphism SL3 Z/2Z ’ G where G is split simple of

(d)

type G2 (i.e., “ A2 ‚ G2 ” ) such that H 1 (F, SL3 Z/2Z) ’ H 1 (F, G) is

surjective.

7. (a) Let Q be a quaternion algebra and let C = C(Q, a) be the Cayley algebra

Q • vQ with v 2 = a. Let AutF (C, Q) be the subgroup of automorphisms

of AutF (C) which map Q to Q. Show that there is an exact sequence

φ

1 ’ SL1 (Q) ’ AutF (C, Q) ’ AutF (Q) ’ 1

’

EXERCISES 507

where φ(y)(a + vb) = a + (vy)b for y ∈ SL1 (Q).

(b) The map SL1 (Q) — SL1 (Q) ’ AutF (C) induced by

(u, x) ’ (a + vb) ’ uau + (vx)(ubu)

is a group homomorphism (i.e., “A2 — A2 ‚ G2 ”).

(Elduque) Let S = (F4 , ) be the unique para-quadratic F2 -algebra. Show

8.

that 1-dimensional algebras and S are the only examples of power-associative

symmetric composition algebras.

9. Let F be a ¬eld of characteristic not 3. Let A be a central simple F -algebra

of degree 3. Compute the quadratic forms TA (x2 ) and SA (x) on A and on A0 ,

and determine their discriminants and their Cli¬ord invariants.

Let » ∈ F — and let (Q, n) be a quaternion algebra. Construct an isomorphism

10.

C(»Q, n), σ M2 (Q), σn⊥n .

Hint: Argue as in the proof of (??).

11. Let (C, , n) be a Cayley algebra and let (C, ) be the associated para-Cayley

algebra, with multiplication x y = x y. Show that

(x a) (a y) = a a (x y) .

(By using the Theorem of Cartan-Chevalley this gives another approach to

triality.)

12. (Elduque) Let C be a Cayley algebra, let (C, ) be the associated para-Cayley

algebra, and let (C• , ) be a Petersson algebra. Let t be a proper similitude

of (C, n), with multiplier µ(t).

(a) If t+ , t’ are such that µ(t)’1 t(x y) = t’ (x) t+ (y), show that

µ(t)’1 t(x y) = •’1 t’ •(x) •t+ •’1 (y).

(b) If θ+ is the automorphism of Spin(C, n) as de¬ned in Proposition (??)

¯

and if θ+ is the corresponding automorphism with respect to ¯, show that

—

¯+ = C(•)θ+ = θ+ C(•).

θ

13. Compute Spin(C, n) for (C, n) a symmetric composition algebra of dimension 2,

resp. 4.

14. Let C be a twisted Hurwitz composition over F — F — F .

(a) If C is a quaternion algebra, show that

AutF (C) = (C — — C — — C — )Det /F — S3

where

(C — — C — — C — )Det = { (a, b, c) ∈ C — | NC (a) = NC (b) = NC (c) }

and S3 acts by permuting the factors.

(b) If C is quadratic,

AutF (C) = SU1 (C) — SU1 (C) (Z/2Z — S3 )

where Z/2Z operates on SU1 (C) — SU1 (C) through (a, b) ’ (a, b) and S3

operates on SU1 (C) — SU1 (C) as in Lemma (??).

15. Describe the action of S3 (triality) on the Weyl group (Z/2Z)3 S4 of a split

simple group of type D4 .

508 VIII. COMPOSITION AND TRIALITY

Notes

§??. The notion of a generic polynomial, which is classical for associative alge-

bras, was extended to strictly power-associative algebras by Jacobson. A systematic

treatment is given in Chap. IV of [?], see also McCrimmon [?].

§??. Octonions (or the algebra of octaves) were discovered by Graves in 1843

and described in letters to Hamilton (see Hamilton [?, Vol. 3, Editor™s Appendix 3,

p. 648]); however Graves did not publish his result and octonions were rediscovered

by Cayley in 1845 [?, I, p. 127, XI, p. 368“371]. Their description as pairs of

quaternions (the “Cayley-Dickson process”) can be found in Dickson [?, p. 15].

Dickson was also the ¬rst to notice that octonions with positive de¬nite norm

function form a division algebra [?, p. 72].

The observation that x(xa) = (xx)a = (ax)x holds in an octonion algebra dates

back to Kirmse [?, p. 76]. The fact that Cayley algebras satisfy the alternative law

was conjectured by E. Artin and proved by Artin™s student Max Zorn in [?]. Artin™s

theorem (that a subalgebra of an alternative algebra generated by two elements is

associative) and the structure theorem (??) ¬rst appeared in [?]. The description of

split octonions as “vector matrices”, as well as the abstract Cayley-Dickson process,

are given in a later paper [?] of Zorn. The fact that the Lie algebra of derivations

of a Cayley algebra is of type G2 and the fact that the group of automorphisms

of the Lie algebra of derivations of a Cayley algebra is isomorphic to the group of

automorphisms of the Cayley algebra if F is a ¬eld of characteristic zero, is given

in Jacobson [?]. In this connection we observe that the Lie algebra of derivations of

the split Cayley algebra over a ¬eld of characteristic 3 has an ideal of dimension 7,

hence is not simple. The fact that the group of automorphisms of a Cayley algebra

is of type G2 is already mentioned without proof by E. Cartan [?, p. 298] [?, p. 433].

Other proofs are found in Freudenthal [?], done by computing the root system, or

in Springer [?], done by computing the dimension of the group and applying the

classi¬cation of simple algebraic groups. In [?] no assumption on the characteristic

of the base ¬eld is made.

Interesting historical information on octonions can be found in the papers of van

der Blij [?] and Veldkamp [?], see also the book of van der Waerden [?, Chap. 10].

The problem of determining all composition algebras has been treated by many

authors (see Jacobson [?] for references). Hurwitz [?] showed that the equation

(x2 + · · · + x2 )(y1 + · · · + yn ) = z1 + · · · + zn

2 2 2 2

1 n

has a solution given by bilinear forms z1 , . . . , zn in the variables x = (x1 , . . . , xn ),

y = (y1 , . . . , yn ) exactly for n = 1, 2, 4, and 8. The determination of all composition

algebras with identity over a ¬eld of characteristic not 2 is due to Jacobson [?]. We

used here the proof of van der Blij-Springer [?], which is also valid in characteristic 2.

A complete classi¬cation of composition algebras (even those without an identity)

is known in dimensions 2 (Petersson [?]) and 4 (Stamp¬‚i-Rollier [?]).

§??. Compositions algebras with associative norms were considered indepen-

dently by Petersson [?], Okubo [?], and Faulkner [?]. We suggest calling them

symmetric composition algebras in view of their very nice (and symmetric) proper-

ties. Applications of these algebras in physics can be found in a recent book [?] by

S. Okubo.

Petersson showed that over an algebraically closed ¬eld symmetric composi-

tions are either para-Hurwitz or, as we call them, Petersson compositions. Okubo

NOTES 509

described para-Cayley Algebras and “split Okubo algebras” as examples of sym-

metric composition algebras. In the paper [?] of Okubo-Osborn it is shown that

over an algebraically closed ¬eld these two types are the only examples of symmetric

composition algebras.

The fact that the trace zero elements in a cubic separable alternative algebra

carry the structure of a symmetric algebra was noticed by Faulkner [?]. The clas-

si¬cation of symmetric compositions, as given in Theorem (??), is due to Elduque-

Myung [?]. However they applied the Zorn Structure Theorem for separable alter-

native algebras, instead of invoking (as we do) the eigenspace decomposition of the

operator e for e an idempotent. The idea to consider such eigenspaces goes back

to Petersson [?]. A similar decomposition for the operator ade is used by Elduque-

Myung in [?]. Connections between the di¬erent constructions of symmetric alge-

bras are clearly described in Elduque-P´rez [?]. We take the opportunity to thank

e

A. Elduque, who detected an error in our ¬rst draft and who communicated [?] to

us before its publication.

Let (A0 , ) be a composition of type 1A2 . It follows from Theorem (??) that

AutF (A0 , ) AutF (A). This can also be viewed in terms of Lie algebras: Since

0

∼

x y’y x = µ(xy’yx), any isomorphism of compositions ± : (A0 , ) ’ (A , ) also

’

0

∼

induces a Lie algebra isomorphism L(A0 ) ’ L(A ). Conversely, (and assuming

’

0

∼

that F has characteristic 0) any isomorphism of Lie algebras L(A0 ) ’ L(A ) ’

∼

extends to an algebra isomorphism A ’ A or the negative of an anti-isomorphism

’

∼

of algebras A ’ A (Jacobson [?, Chap. X, Theorem 10]). However the negative of

’

an anti-isomorphism of algebras cannot restrict to an isomorphism of composition

algebras. In particular we see that AutF (A0 , ) is isomorphic to the connected

component AutF L(A0 ) 0 of AutF L(A0 ) .

§??. We introduce triality using symmetric composition algebras of dimen-

sion 8 and their Cli¬ord algebras. Most of the results for compositions of type G2

can already be found in van der Blij-Springer [?], Springer [?], Wonenburger [?], or

Jacobson [?, p. 78], [?]. However the presentation through Cli¬ord algebras given

here, which goes back to [?], is di¬erent. The use of symmetric compositions also

has the advantage of giving very symmetric formulas for triality. The isomorphism

∼

of algebras C(S, n) ’ EndF (S • S) for symmetric compositions of dimension 8

’

can already be found in the paper [?] of Okubo and Myung. A di¬erent approach

to triality can be found in the book of Chevalley [?].

Triality in relation to Lie groups is discussed brie¬‚y by E. Cartan [?, Vol. II,

1

§139] as an operation permuting the vector and the 2 -spinor representations of D4 .

The ¬rst systematic treatment is given in Freudenthal [?], where local triality (for

Lie algebras) and global triality is discussed.

There is also an (older) geometric notion of triality between points and spaces of

two kinds on a (complex) 6-dimensional quadric in P7 . These spaces correspond to

maximal isotropic spaces of the quadric given by the norm of octonions. Geometric

triality goes back to Study [?] and E. Cartan [?, pp. 369-370], see also [?, I, pp. 563“

565]; A systematic study of geometric triality is given in Vaney [?], Weiss [?], see

also Kuiper [?]. Geometric applications can be found in the book on “Punktreihen-

geometrie” of Weiss [?].

The connection between triality and octonions, already noticed by Cartan,

is used systematically by Vaney and Weiss. The existence of triality is, in fact,

“responsible” for the existence of Cayley algebras (see Tits [?]). A systematic

510 VIII. COMPOSITION AND TRIALITY

description of triality in projective geometry in relation to the theory of groups is

given in Tits [?].

The paper of van der Blij-Springer [?] gives a very nice introduction to triality

in algebra and geometry. There is also another survey article, by Adams [?].

§??. The notion of a twisted composition (due to Rost) was suggested by the

construction of cyclic compositions, due to Springer [?]. Many results of this section,

for example Theorem (??), were inspired by the notes [?].

CHAPTER IX

Cubic Jordan Algebras

The set of symmetric elements in an associative algebra with involution admits

the structure of a Jordan algebra. One aim of this chapter is to give some insight

into the relationship between involutions on central simple algebras and Jordan

algebras. After a short survey on central simple Jordan algebras in §??, we spe-

cialize to Jordan algebras of degree 3 in §??; in particular, we discuss extensively

“Freudenthal algebras,” a class of Jordan algebras connected with Hurwitz algebras

and we describe the Springer construction, which ties twisted compositions with cu-

bic Jordan algebras. On the other hand, cubic Jordan algebras are also related to

cubic associative algebras through the Tits constructions (§??). Of special interest,

and the main object of study of this chapter, are the exceptional simple Jordan al-

gebras of dimension 27, whose automorphism groups are of type F4 . The di¬erent

constructions mentioned above are related to interesting subgroups of F4 . For ex-

ample, the automorphism group of a split twisted composition is a subgroup of F4

and outer actions on Spin8 (triality!) become inner over F4 . Tits constructions

are related to the action of the cyclic group Z/3Z on Spin 8 which yields invariant

subgroups of classical type A2 , and Freudenthal algebras are related to the action

of the group S3 on Spin8 which yields invariant subgroups of exceptional type G2 .

Cohomological invariants of exceptional simple Jordan algebras are discussed

in the last section.

§37. Jordan algebras

We assume in this section that F is a ¬eld of characteristic di¬erent from 2. A

Jordan algebra J is a commutative ¬nite dimensional unitary F -algebra such that

the multiplication (a, b) ’ a q b satis¬es

(37.1) (a q a) q b q a = (a q a)(b q a)

for all a, b ∈ J. For any associative algebra A, the product

a q b = 1 (ab + ba)

2

gives A the structure of a Jordan algebra, which we write A+ . If B is an associative

algebra with involution „ , the set Sym(B, „ ) of symmetric elements is a Jordan

subalgebra of B + which we denote H(B, „ ).

Observe that A+ H(B, „ ) if B = A — Aop and „ is the exchange involution.

A Jordan algebra A is special if there exists an injective homomorphism A ’

+

D for some associative algebra D and is exceptional otherwise.

A Jordan algebra is strictly power-associative and we write an for the nth power

of an element a. Hence it admits a generic minimal polynomial

PJ,x (X) = X m ’ s1 (x)X m’1 + · · · + (’1)m sm (x)1,

511

512 IX. CUBIC JORDAN ALGEBRAS

where TJ = s1 is the generic trace and NJ = sm the generic norm. The bilinear

trace form T (x, y) = TJ (xy) is associative (see Corollary (??)). By Dieudonn´™s

e

theorem (??), a Jordan algebra is separable if T is nonsingular. The converse is a

consequence of the following structure theorem:

(37.2) Theorem. (1) Any separable Jordan F -algebra is the product of simple

Jordan algebras whose centers are separable ¬eld extensions of F .

(2) A central simple Jordan algebra is either

(a) the Jordan algebra of a nondegenerate quadratic space of dimension ≥ 2,

(b) a Jordan algebra H(B, „ ) where B is associative and K-central simple as

an algebra with involution „ , and where K is either quadratic ´tale and „

e

is unitary with respect to K or K = F and „ is F -linear, or

(c) an exceptional Jordan algebra of dimension 27.

Reference: (??) is [?, Theorem 4, p. 239], and (??) (which goes back to Albert [?])

follows from [?, Corollary 2, p. 204] and [?, Theorem 11, p. 210]. We de¬ne and

discuss the di¬erent types occurring in (??) in the following sections.

Let Sepjord n (m) be the groupoid of separable Jordan F -algebras of dimension n

and degree m with isomorphisms as morphisms.

37.A. Jordan algebras of quadratic forms. Let (V, q) be a nonsingular

¬nite dimensional quadratic space with polar bq (x, y) = q(x + y) ’ q(x) ’ q(y). We

de¬ne a multiplication on J(V, q) = F • V by setting

1

(», v) q (µ, w) = »µ + 2 bq (v, w), »w + µv

for v, w ∈ V and », µ ∈ F . The element (1, 0) is an identity and the canonical

embedding of J(V, q) = F q 1•V into the Cli¬ord algebra C(V, q) shows that J(V, q)

is a Jordan algebra (and is special). The generic minimal polynomial of J(V, q) is

PJ,a (X) = X 2 ’ 2ξX + ξ 2 ’ q(v) 1

where a = (ξ, v) ∈ F q 1 • V , hence J(V, q) has degree 2, the trace is given by

TJ (ξ, v) = 2ξ and the norm by NJ (ξ, v) = ξ 2 ’ q(v). Thus NJ is a nonsingular

quadratic form. The bilinear trace form T : (x, y) ’ TJ (x q y) is isomorphic to

2 ⊥ bq , furthermore T is associative, hence by (??) J is separable if and only if

q is nonsingular. We set J : Qn ’ Sepjord n+1 (2) for the functor (V, q) ’ J(V, q).

Let J be a separable Jordan algebra of degree 2, with generic minimal polynomial

PJ,a (X) = X 2 ’ TJ (a)X + NJ (a)1.

Linearizing and taking traces shows that

2TJ (x q y) ’ 2TJ (x)TJ (y) + 2bNJ (x, y) = 0,

with bNJ the polar of NJ ; hence

(37.3) bNJ (x, y) = TJ (x)TJ (y) ’ TJ (x q y).

For J 0 = { x ∈ J | TJ (x) = 0 }, we have an orthogonal decomposition

J = F q 1 ⊥ J0

with respect to the bilinear trace form T as well as with respect to NJ and, in view

of (??), NJ is nonsingular on J 0 if and only if T is nonsingular on J 0 if and only

if J is separable. Let

Q : Sepjord n+1 (2) ’ Qn

§37. JORDAN ALGEBRAS 513

be the functor given by J ’ (J 0 , ’NJ ).

(37.4) Proposition. The functors J and Q de¬ne an equivalence of groupoids

Qn ≡ Sepjord n+1 (2).

In particular we have Autalg J(V, q) = O(V, q), so that Jordan algebras of type

Sepjord n+1 (2) are classi¬ed by H 1 (F, On ).

Proof : The claim follows easily from the explicit de¬nitions of J and Q.

(37.5) Remark. If dim V ≥ 2, J(V, q) is a simple√ Jordan algebra. If dim V = 1,

J(V, q) is isomorphic to the quadratic algebra F ( ») = F (X)/(X 2 ’ ») where

q ».

We next consider Jordan algebras of degree ≥ 3 and begin with Jordan algebras

associated to central simple algebras with involution.

37.B. Jordan algebras of classical type. Let K be an ´tale quadratic

e

algebra over F with conjugation ι or let K = F and ι = 1. Let (B, „ ) be a

K-central simple algebra with „ an ι-linear involution. As in Chapter ?? we denote

the groupoids corresponding to di¬erent types of involutions by An , Bn , Cn , and Dn .

We set A+ , Bn , Cn , resp. Dn for the groupoids of Jordan algebras whose objects

+ + +

n

are sets of symmetric elements H(B, „ ) for (B, „ ) ∈ A, B, C , resp. D. For each of

these categories A, B, C , D, we have functors S : A ’ A+ , . . . , D ’ D + induced

by (B, „ ) ’ H(B, „ ).

(37.6) Proposition. Let B, B be K-central simple with involutions „ , „ , of de-

∼

gree ≥ 3. Any isomorphism H(B, „ ) ’ H(B , „ ) of Jordan algebras extends to a

’

∼

unique isomorphism (B, „ ) ’ (B , „ ) of K-algebras with involution. In particular

’

H(B, „ ) and H(B , „ ) are isomorphic Jordan algebras if and only if (B, „ ) and

(B , „ ) are isomorphic as K-algebras with involution and the functor S induces an

isomorphism of corresponding groupoids.

Reference: See Jacobson [?, Chap. V, Theorem 11, p. 210].

Thus, in view of Theorem (??), the classi¬cation of special central simple

Jordan algebras of degree ≥ 3 is equivalent to the classi¬cation of central simple

associative algebras with involution of degree ≥ 3.

If (B, „ ) is a central simple algebra with a unitary involution over K, we have

an exact sequence of group schemes

Z/2Z) ’ 1

1 ’ AutK (B, „ ) ’ Aut(B, „ ) ’ Autalg (K)(

Thus there is a sequence

1 ’ AutK (B, „ ) ’ Aut H(B, „ ) ’ Z/2Z ’ 1.

If B = A — Aop and „ is the exchange involution, we obtain

1 ’ Aut(A) ’ Aut(A+ ) ’ Z/2Z ’ 1

(37.7)

and the sequence splits if A admits an anti-automorphism. The group scheme

Aut H(B, „ ) is smooth in view of Proposition (??), (??), since its connected

component PGU(B, „ ) = AutK (B, „ ) is smooth. Thus Aut(A+ ) is smooth too.

514 IX. CUBIC JORDAN ALGEBRAS

37.C. Freudenthal algebras. Let C be a Hurwitz algebra with norm NC

and trace TC over a ¬eld F of characteristic not 2 and let

Mn (C) = Mn (F ) — C.

For X = (cij ) ∈ Mn (C), let X = (cij ) where c ’ c, c ∈ C, is conjugation. Let

¯

± = diag(±1 , ±2 , . . . , ±n ) ∈ GLn (F ). Let

t

Hn (C, ±) = { X ∈ Mn (C) | ±’1 X ± = X }.

Let n ≥ 3. If C is associative, Hn (C, ±) and twisted forms of Hn (C, ±) are Jordan

1

algebras of classical type for the product X q Y = 2 (XY + Y X) where XY is the

usual matrix product. In particular they are special. If n = 3 and C = C is a Cayley

algebra, H3 (C, ±) (and twisted forms of H3 (C, ±)) are Jordan algebras for the same

multiplication (see for example Jacobson [?, Chap. III, Theorem 1, p. 127]). For

n = 2 we still get Jordan algebras since H2 (C, ±) can be viewed as a subalgebra

of H3 (C, ±) with respect to a Peirce decomposition ([?, Chap. III, Sect. 1]) relative

to the idempotent diag(1, 0, 0). In fact, H2 (C, ±) is, for any Hurwitz algebra C,

separable of degree 2 hence special (see Exercise 3). However the algebra H 3 (C, ±)

and twisted forms of H3 (C, ±) are exceptional Jordan algebras (Albert [?]). In fact

they are not even homomorphic images of special Jordan algebras (Albert-Paige

[?] or Jacobson [?, Chap. I, Sect. 11, Theorem 11]). Conversely, any central simple

exceptional Jordan algebra is a twisted form of H3 (C, ±) for some Cayley algebra

C (Albert [?, Theorem 17]).

The elements of J = H3 (C, ±) can be represented as matrices

«

±’1 ±3 c2

ξ1 c3 ¯

1

a = ±’1 ±1 c3 c1 , ci ∈ C, ξi ∈ F

(37.8) ¯ ξ2

2

’1

c2 ±3 ±2 c 1

¯ ξ3

and the generic minimal polynomial is (Jacobson [?, p. 233]):

PJ,a (X) = X 3 ’ TJ (a)X 2 + SJ (a)X ’ NJ (a)1

where

TJ (a) = ξ1 + ξ2 + ξ3 ,

SJ (a) = ξ1 ξ2 + ξ2 ξ3 + ξ1 ξ3 ’ ±’1 ±2 NC (c1 ) ’ ±’1 ±3 NC (c2 ) ’ ±’1 ±1 NC (c3 ),

3 1 2

NJ (a) = ξ1 ξ2 ξ3 ’ ±’1 ±2 ξ1 NC (c1 ) ’ ±’1 ±3 ξ2 NC (c2 ) ’ ±’1 ±1 ξ3 NC (c3 )

3 1 2

+ TC (c3 c1 c2 ).

Let

«

¯

±’1 ±3 d2

·1 d3 1

¯

b = ± 2 ± 1 d 3 d1 ,

’1

di ∈ C, ·i ∈ F.

·2

¯

’1

d2 ±3 ±2 d 1 ·3

Let bC be the polar of NC . The bilinear trace form T : (a, b) ’ TJ (a q b) is given by

T (a, b) =

ξ1 ·1 + ξ2 ·2 + ξ3 ·3 + ±’1 ±2 bC (c1 , d1 ) + ±’1 ±3 bC (c2 , d2 ) + ±’1 ±1 bC (c3 , d3 )

3 1 2

or

T = 1, 1, 1 ⊥ bC — ±’1 ±2 , ±’1 ±3 , ±’1 ±1 ,

(37.9) 3 1 2

§37. JORDAN ALGEBRAS 515

Thus T is nonsingular. The quadratic form SJ is the quadratic trace which is a

regular quadratic form. Furthermore one can check that

(37.10) bSJ (a, b) = TJ (a)TJ (b) ’ TJ (a q b).

We have T (1, 1) = 3; hence there exists an orthogonal decomposition

H3 (C, ±) = F · 1 ⊥ H3 (C, ±)0 , H3 (C, ±)0 = { x ∈ H3 (C, ±) | TJ (x) = 0 }

if char F = 3.

We call Jordan algebras isomorphic to algebras H3 (C, ±), for some Hurwitz

algebra C, reduced Freudenthal algebras and we call twisted forms of H 3 (C, ±)

Freudenthal algebras. If we allow C to be 0 in H3 (C, ±), the split cubic ´tale algebra

e

F — F — F can also be viewed as a special case of a reduced Freudenthal algebra.

Hence cubic ´tale algebras are Freudenthal algebras of dimension 3. Furthermore, if

e

char F = 3, it is convenient to view F as a Freudenthal algebra with norm NF (x) =

x3 . Freudenthal algebras H3 (C, s), with C = 0 or C a split Hurwitz algebra and

s = diag(1, ’1, 1) are called split. A Freudenthal algebra can have dimension 1, 3,

6, 9, 15, or 27. In dimension 3 Freudenthal algebras are commutative cubic ´tale F -

e

algebras and in dimension greater than 3 central simple Jordan algebras of degree 3

over F . The group scheme G of F -automorphisms of the split Freudenthal algebra

of dimension 27 is simple exceptional split of type F4 (see Theorem (??)). Since

the ¬eld extension functor j : F4 (F ) ’ F4 (Fsep ) is a “-embedding (see the proof of

Theorem (??)) Freudenthal algebras of dimension 27 (which are also called Albert

algebras) are classi¬ed by H 1 (F, G):

(37.11) Proposition. Let G be a simple split group of type F4 . Albert algebras (=

simple exceptional Jordan algebras of dimension 27) are classi¬ed by H 1 (F, G).

It is convenient to distinguish between Freudenthal algebras with zero divisors

and Freudenthal algebras without zero divisors (“division algebras”).

(37.12) Theorem. Let J be a Freudenthal algebra.

(1) If J has zero divisors, then J F — K, K a quadratic ´tale F -algebra, if

e

dimF J = 3, and J H3 (C, ±) for some Hurwitz algebra C, i.e., J is reduced if

dimF J > 3. Moreover, a Freudenthal algebra J of degree > 3 is reduced if and

only if J contains a split ´tale algebra L = F — F — F . More precisely, if e i ,

e

i = 1, 2, 3, are primitive idempotents generating L, then there exist a Hurwitz

algebra C, a diagonal matrix ± = diag(±1 , ±2 , ±3 ) ∈ GL3 (F ) and an isomorphism

∼

φ : J ’ H3 (C, ±) such that φ(ei ) = Eii .

’

(2) If J does not have zero divisors, then either J = F + (if char F = 3), J = L+ for

a cubic (separable) ¬eld extension L of F , J = D + for a central division algebra D,

J = H(B, „ ) for a central division algebra B of degree 3 over a quadratic ¬eld

extension K of F and „ a unitary involution or J is an exceptional Jordan division

algebra of dimension 27 over F .

Reference: The ¬rst part of (??) and the last claim of (??) follow from the clas-

si¬cation theorem (??) and the fact, due to Schafer [?], that Albert algebras with

zero divisors are of the form H3 (C, ±). The last claim in (??) is a special case of

the coordinatization theorem of Jacobson [?, Theorem 5.4.2].

In view of a deep result of Springer [?, Theorem 1, p. 421], the bilinear trace

form is an important invariant for reduced Freudenthal algebras. The result was

generalized by Serre [?, Th´or`me 10] and Rost as follows:

ee

516 IX. CUBIC JORDAN ALGEBRAS

(37.13) Theorem. Let F be a ¬eld of characteristic not 2. Let J, J be reduced

Freudenthal algebras. Let T , resp. T , be the corresponding bilinear trace forms.

The following conditions are equivalent:

(1) J and J are isomorphic.

(2) T and T are isometric.

Furthermore, if (??) (or (??)) holds, J H3 (C, ±) and J H3 (C , ± ), then C

and C are isomorphic.

Proof : We may assume that J = H3 (C, ±) and J = H3 (C , ± ) with C, C = 0.

(??) implies (??) by uniqueness of the generic minimal polynomial. Assume now

that T and T are isometric. The bilinear trace of H3 (C, ±) is of the form

T = 1, 1, 1 ⊥ bC — ±’1 ±2 , ±’1 ±3 , ±’1 ±1

3 1 2

and a similar formula holds for T . Thus

’1 ’1 ’1

bC — ±’1 ±2 , ±’1 ±3 , ±’1 ±1

(37.14) bC — ±3 ±2 , ±1 ±3 , ±2 ±1 .

3 1 2

We show in the following Lemma (??) that (??) implies NC NC , hence C

C holds by Proposition (??), and we may identify C and C . Assume next

that C is associative. By Jacobson [?], (??) implies that the C-hermitian forms

’1 ’1 ’1

±’1 ±2 , ±’1 ±3 , ±’1 ±1 C and ±3 ±2 , ±1 ±3 , ±2 ±1 C are isometric. They are

3 1 2

similar to ±1 , ±2 , ±3 C , resp. ±1 , ±2 , ±3 C . Thus ±, ± de¬ne isomorphic unitary

involutions on M3 (C) and the Jordan algebras H3 (C, ±) and H3 (C, ± ) are isomor-

phic. If C is a Cayley algebra, the claim is much deeper and we need Springer™s

result, which says that H3 (C, ±) and H3 (C, ± ) are isomorphic if their trace forms

are isometric (Springer [?, Theorem 1, p. 421]), to ¬nish the proof.

For any P¬ster form •, let • = 1⊥ .

(37.15) Lemma. Let φn , ψn be n-P¬ster bilinear forms and χp , •p p-P¬ster bi-

linear forms for p ≥ 2. If φn — χp ψn — •p , then φn ψn and φn — χp ψn — •p .

Proof : We make computations in the Witt ring W F and use the same notation

for a quadratic form and its class in W F . Let q = φn — χp = ψn — •p . Adding

φn , resp. ψn on both sides , we get that q + φn and q + ψn lie in I n+p F , so that

ψn ’ φn ∈ I n+p F . Since ψn ’ φn can be represented by a form of rank 2n+1 ’ 2,

it follows from the Arason-P¬ster Hauptsatz (Lam [?, Theorem 3.1, p. 289]), that

ψn ’ φn = 0.

(37.16) Corollary. Let T = 1, 1, 1 ⊥ bNC — ’b, ’c, bc be the trace form of

J = H3 (C, ±) and let q be the bilinear P¬ster form bNC — b, c . The isometry

class of T determines the isometry classes of NC and q. Conversely, the classes of

NC and q determine the class of T .

Proof : The claim is a special case of Lemma (??).

(37.17) Remark. Theorem (??) holds more generally for separable Jordan alge-

bras of degree 3: In view of the structure theorem (??) the only cases left are

algebras of the type F — J(V, q), where the claim follows from Proposition (??),

and ´tale algebras of dimension 3 with zero divisors. Here the claim follows from

e

the fact that quadratic ´tale algebras are isomorphic if and only if their norms are

e

isomorphic (see Proposition (??)).

An immediate consequence of Theorem (??) is:

§38. CUBIC JORDAN ALGEBRAS 517

(37.18) Corollary. H3 (C, ±) is split for any ± if C is split.

(37.19) Remark. Conditions on ±, ± so that H3 (C, ±) and H3 (C, ± ) are isomor-

phic for a Cayley division algebra C are given in Albert-Jacobson [?, Theorem 5].

We conclude this section with a useful “Skolem-Noether” theorem for Albert

algebras:

(37.20) Proposition. Let I, I be reduced simple Freudenthal subalgebras of de-

∼

gree 3 of a reduced Albert algebra J. Any isomorphism φ : I ’ I can be extended

’

to an automorphism of J.

Reference: See Jacobson [?, Theorem 3, p. 370].

However, for example, split cubic ´tale subalgebras of a reduced Albert al-

e

gebra J are not necessarily conjugate by an automorphism of J. Necessary and

su¬cient conditions are given in Albert-Jacobson [?, Theorem 9]. It would be in-

teresting to have a corresponding result for a pair of arbitrary isomorphic cubic

´tale subalgebras.

e

Another Skolem-Noether type of theorem for Albert algebras is given in (??).

§38. Cubic Jordan Algebras

A separable Jordan algebra of degree 3 is either a Freudenthal algebra or is of

the form F + — J(V, q) where J(V, q) is the Jordan algebra of a quadratic space of

dimension ≥ 2 (see the structure theorem (??)); if J is a Freudenthal algebra, then

J is of the form F + (assuming char F = 3), L+ for L cubic ´tale, classical of type

e

A2 , B1 , C3 or exceptional of dimension 27. Let

PJ,a (X) = X 3 ’ TJ (a)X 2 + SJ (a)X ’ NJ (a)1

be the generic minimal polynomial of a separable Jordan algebra J of degree 3.

The element

x# = x2 ’ TJ (x)x + SJ (x)1 ∈ J

obviously satis¬es x q x# = NJ (x)1. It is the (Freudenthal ) adjoint of x and the

linearization of the quadratic map x ’ x#

x — y = (x + y)# ’ x# ’ y #

= 2x q y ’ TJ (x)y ’ TJ (y)x + bSJ (x, y)1

is the Freudenthal “—”-product.34 Let T (x, y) = TJ (x q y) be the bilinear trace

form. The datum (J, NJ , #, T, 1) has the following properties (see McCrimmon [?,

Section 1]):

(a) the form NJ : J ’ F is cubic, the adjoint # : J ’ J, x ’ x# , is a quadratic

map such that x## = N (x)x and 1 ∈ J is a base point such that 1# = 1;

(b) the nonsingular bilinear trace form T is such that

NJ (x + »y) = »3 NJ (y) + »2 T (x# , y) + »T (x, y # ) + NJ (x)

and T (x, 1)1 = 1 — x + x for x, y ∈ J and » ∈ F .

1

34 The (x + y)# ’ x# ’ y # .

—-product is sometimes de¬ned as 2

518 IX. CUBIC JORDAN ALGEBRAS

These properties are characteristic-free. Following McCrimmon [?] and Petersson-

Racine [?] (see also Jacobson [?, 2.4]), we de¬ne a cubic norm structure over any

¬eld F (even if char F = 2) as a datum (J, N, #, T, 1) with properties (??) and (??).

An isomorphism

∼

φ : (J, N, #, T, 1) ’ (J , N , #, T , 1 )

’

∼ ∼

is an F -isomorphism J ’ J of vector spaces which is an isometry (J, N, T ) ’

’ ’

(J , N , T ), such that φ(1) = 1 and φ(x# ) = φ(x)# for all x ∈ J. We write Cubjord

for the groupoid of cubic norm structures with isomorphisms as morphisms.

(38.1) Examples. Forgetting the Jordan multiplication and just considering ge-

neric minimal polynomials, we get cubic norm structures on J = H3 (C, ±) and on

twisted forms of these. If J = L is cubic ´tale over F , NL = NL/F , TL = TL/F ,

e

and T is the trace form. If J is of classical type A, B, or D, then NJ is the

reduced norm and TJ is the reduced trace. If J is of classical type C , then NJ

is the reduced pfa¬an and TJ is the reduced pfa¬an trace. We also have cu-

bic structures associated to quadratic forms, as in the case of the Jordan algebra

J = F + — J(V, q). More generally, let J = (V , q , 1 ) be a pointed quadratic

space, i.e., 1 ∈ V is such that q (1 ) = 1, and let b be the polar of q . On

J = F • J we de¬ne NJ (x, v) = xq (v), 1 = (1, 1 ), TJ (x, v) = x + b (1 , v),

T (x, v), (y, w) = xy + b (v, w) where w = b (1 , w)1 ’ w and (x, v)# = q (v), xv .

Conversely, any cubic norm structure is of one of the types described above, see

for example Petersson-Racine [?, Theorem 1.1]. We refer to cubic norm structures

associated with Freudenthal algebras as Freudenthal algebras (even if they do not

necessarily admit a multiplication!). Cubic norm structures of the form H3 (C, ±)

for arbitrary C and ± are called reduced Freudenthal algebras.

(38.2) Lemma. Let (J, N, #, T, 1) be a cubic norm structure and set

x2 = T (x, 1)x ’ x# — 1 x3 = T (x, x)x ’ x# — x.

and

(1) Any element x ∈ J satis¬es the cubic equation

P (x) = x3 ’ TJ (x)x2 + SJ (x)x ’ NJ (x)1 = 0

where TJ (x) = T (x, 1) and SJ (x) = TJ (x# ). Furthermore we have

x# = x2 ’ TJ (x)x + SJ (x)1.

In particular any element x ∈ J generates a commutative associative cubic unital

algebra F [x] ‚ J.

(2) There is a Zariski-open, non-empty subset U of J such that F [x] is ´tale for

e

x ∈ U.

(3) The identities

(a) SJ (1) = TJ (1) = 3, NJ (1) = 1, 1# = 1,

(b) SJ (x) = TJ (x# ), bSJ (x, y) = TJ (x — y),

(c) bSJ (x, 1) = 2TJ (x),

(d) 2SJ (x) = TJ (x)2 ’ TJ (x2 ),

(e) TJ (x — y) = TJ (x)TJ (y) ’ T (x, y),

(f) x## = NJ (x)x,

(g) T (x — y, z) = T (x, y — z)

hold in J.

§38. CUBIC JORDAN ALGEBRAS 519

Proof : (??) can be directly checked. For (??) we observe that F [x] is ´tale if and

e

only if the generic minimal polynomial PJ,x of x has pairwise distinct roots (in an

algebraic closure) i.e., the discriminant of PJ,x (as a function of x) is not zero. This

de¬nes the open set U . It can be explicitely shown that the set U is non-empty

if J is reduced, i.e., is not a division algebra. Thus we may assume that J is a

division algebra. Then, by the following lemma (??), F is in¬nite. Again by (??)

J is reduced over an algebraic closure Falg of F . The set U being non-empty over

Falg and F being in¬nite, it follows that U is non-empty. We refer to [?] or [?]

for (??). A proof for cubic alternative algebras is in (??).

An element x ∈ J is invertible if it is invertible in the algebra F [x] ‚ J.

We say that a cubic norm structure is a division cubic norm structure if every

nonzero element has an inverse. Such structures are (non-reduced) Freudenthal

algebras and can only exist in dimensions 1, 3, 9, and 27. In dimension 3 we get

separable ¬eld extensions and in dimension 9 central associative division algebras of

degree 3 or symmetric elements in central associative division algebras of degree 3

over quadratic separable ¬eld extensions, with unitary involutions. Corresponding

examples in dimension 27 will be given later using Tits constructions.

(38.3) Lemma. An element x ∈ J is invertible if and only if NJ (x) = 0 in F . In

that case we have x’1 = NJ (x)’1 x# . Thus a cubic norm structure J is a division

cubic norm structure if and only if NJ (x) = 0 for x = 0 in J, i.e., NJ is anisotropic.

In particular a cubic norm structure J of dimension > 3 is reduced (i.e., is not a

division algebra) if F is ¬nite or algebraically closed.

Proof : If NJ (x) = 0 for x = 0, we have by Lemma (??) x## = NJ (x)x = 0 hence

either u = x# or u = x satis¬es u# = 0 and u = 0. We then have SJ (u) =

TJ (u# ) = 0 so that u satis¬es

0 = u# = u2 ’ TJ (u)u = 0.

If TJ (u) = 0 we have u2 = 0; if TJ (u) = 0 we may assume that TJ (u) = 1 and

u2 = u, however u = 1. Thus in both cases u is not invertible (see also Exercise ??

of this chapter). The claim for F ¬nite or algebraically closed follows from the fact

that such a ¬eld is Ci , i ¤ 1 (see for example the book of Greenberg [?, Chap. 2]

or Scharlau [?, § 2.15]). Thus NJ , which is a form of degree 3 in 9 or 27 variables

cannot be anisotropic over a ¬nite ¬eld or an algebraically closed ¬eld.

(38.4) Proposition. If char F = 2, the categories Cubjord and Sepjord(3) are

isomorphic.

Proof : Any separable cubic Jordan algebra determines a cubic norm structure and

an isomorphism of separable cubic Jordan algebras is an isomorphism of the corre-

sponding structures. Conversely,

x q y = 2 [(x + y)2 ’ x2 ’ y 2 ]

1

de¬nes on the underlying vector space J of a cubic norm structure J a Jordan

multiplication and an isomorphism of cubic norm structures is an isomorphism for

this multiplication.

520 IX. CUBIC JORDAN ALGEBRAS

38.A. The Springer Decomposition. Let L = F [x] be a cubic ´tale subal-

e

gebra of a Freudenthal algebra J (in the new sense). Since L is ´tale, the bilinear

e

trace form TJ |L is nonsingular and there is an orthogonal decomposition

V = L⊥ ‚ J 0 = { x ∈ J | TJ (x) = 0 }.

J =L⊥V with

We have TJ ( ) = T ( , 1) = TL/F ( ) and NJ ( ) = NL/F ( ) for ∈ L. It follows from

T ( 1 — 2 , v) = T ( 1 , 2 — v) = 0 for v ∈ V that — v ∈ V for ∈ L and v ∈ V . We

de¬ne

—¦ v = ’ — v,

so that —¦ v ∈ V for ∈ L and v ∈ V . Further, let Q : V ’ L and β : V ’ V be

the quadratic maps de¬ned by setting

v # = ’Q(v), β(v) ∈ L • V,

so that

( , v)# = #

’ Q(v), β(v) ’ —¦ v .

We have

SJ (v) = TJ (v # ) = ’TL/F Q(v)

since T β(v) = 0. Furthermore, putting β(v, w) = β(v + w) ’ β(v) ’ β(w), we get

v — w = ’bQ (v, w), β(v, w) .

(38.5) Example. Let J = H3 (C, 1) be a reduced Freudenthal algebra and let

L = F — F — F ‚ J be the set of diagonal elements. Then V is the space of

matrices

«

0 c 3 c2

¯

v = c 3 0 c 1 , c i ∈ C

¯

c 2 c1 0

¯

and the “—¦”-action of L on V is given by

« «

0 c3

¯ c2 0 »3 c 3

¯ »2 c 2

c 3 0 = »3 c3 »1 c 1

c1

¯ 0 ¯

(»1 , »2 , »3 ) —¦

c 2 c1

¯ 0 »2 c 2

¯ »1 c 1 0

Identifying C • C • C with V through the map

«

0 c3

¯ c2

v = (c1 , c2 , c3 ) ’ c3 0 c1

¯

c 2 c1

¯ 0

the action of L on V is diagonal, hence V is an L-module. We have

Q(v) = (c1 c1 , c2 c2 , c3 c3 )

for v = (c1 , c2 , c3 ), so that (V, Q) is a quadratic space over L. Furthermore we get

β(v) = (c2 c3 , c3 c1 , c1 c2 ),

hence β( —¦ v) = # —¦ β(v), Q β(v) = Q(v)# , and bQ v, β(v) = NJ (v) ∈ F . Thus

(V, L, Q, β) is a twisted composition.

The properties of the “—¦”-action described in Example (??) hold in general:

§38. CUBIC JORDAN ALGEBRAS 521

(38.6) Theorem. (1) Let (J, N, #, T, 1) be a cubic Freudenthal algebra, let L be a

cubic ´tale subalgebra of dimension 3 and let V = L⊥ for the bilinear trace form T .

e

The operation L—V ’ V given by ( , v) ’ —¦v de¬nes the structure of an L-module

on V such that (L, Q) is a quadratic space and (V, L, Q, β) is a twisted composition.

(2) For any twisted composition (V, L, Q, β), the cubic structure (J, N, #, T, 1) on

the vector space J(L, V ) = L • V given by

N ( , v) = NL/F ( ) + bQ v, β(v) ’ TL Q(v) ,

( , v)# = #

’ Q(v), β(v) ’ —¦ v ∈ L • V

T ( 1 , v1 ), ( 2 , v2 ) = TL/F ( 1 2) + TL/F bQ (v1 , v2 )

is a Freudenthal algebra. Furthermore we have

SJ ( , v) = TJ ( , v)# = TJ ( #

) ’ TJ Q(v) .

Proof : (??) It su¬ces to check that V is an L-module over a separable closure,

and there we may assume by (??) (which also holds for Freudenthal algebras in

the new sense) that L = F — F — F is diagonal in some H3 (C, ±). The claim

then follows from Example (??). Claim (??) can also be checked rationally, see

Petersson-Racine [?, Proposition 2.1] (or Springer [?], if char F = 2).

(??) If L is split and V = C • C • C, we may identify J(V, L) with H3 (C, 1)

as in Example (??). The general case then follows by descent.

We say that the Freudenthal algebra J(V, L) is the Springer construction asso-

ciated with the twisted composition (V, L, Q, β). This construction was introduced

by Springer for cyclic compositions of dimension 8, in relation to exceptional Jordan

algebras. Conversely, given L ‚ J ´tale of dimension 3, we get a Springer decom-

e

position J = L • V . Springer decompositions for arbitrary cubic structures were

¬rst considered by Petersson-Racine [?, Section 2]. Any Freudenthal algebra is (in

many ways) a Springer construction.

Let “s = (V, L, Q, β) be a split twisted composition of dimension 8. Its associ-

ated algebraic group of automorphisms is Spin8 S3 (see (??)). The corresponding

Freudenthal algebra Js = J(V, L) is split; we recall that by Theorem (??) its auto-

morphism group de¬nes a simple split algebraic group G of type F4 .

(38.7) Corollary. The map “s ’ Js induces an injective group homomorphism

S3 ’ G. The corresponding map in cohomology H 1 (F, Spin8 S3 ) ’

Spin8

H 1 (F, G), which associates the class of J(V, L) to a twisted composition (V, L, Q, β),

is surjective.

Proof : Let (V, L, Q, β) be a twisted split composition of dimension 8. Clearly any

automorphism of (V, L, Q, β) extends to an automorphism of J(L, V ) and conversely

any automorphism of J(L, V ) which maps L to L restricts to an automorphism of

(V, L, Q, β). This shows the injectivity of Spin8 S3 ’ G. The second claim follows

from the facts that H 1 (F, Spin8 S3 ) classi¬es twisted compositions of dimension 8

(Proposition (??)), that H 1 (F, G) classi¬es Albert algebras (Proposition (??)) and

that any Albert algebra admits a Springer decomposition.

(38.8) Theorem. Let J(V, L) be the Springer construction associated with a twis-

ted composition (V, L, Q, β). Then J(V, L) has zero divisors if and only if the

twisted composition (V, L, Q, β) is similar to a Hurwitz composition “(C, L) for

some Hurwitz algebra C. Furthermore, we have J(V, L) H3 (C, ±) for some ±

(and the same Hurwitz algebra C).

522 IX. CUBIC JORDAN ALGEBRAS

Proof : By Theorem (??) the composition (V, L, Q, β) is similar to a Hurwitz compo-

sition “(C, L) if and only if there exists v ∈ V such that β(v) = »v and N (v) = »#

for some » = 0 ∈ L; we then have (v, »)# = 0 in J(V, L). By Exercise ?? of this

chapter, this is equivalent with the existence of zero divisors in J(V, L) (see also the

proof of (??)). Hence Theorem (??) implies that J(V, L) is a reduced Freudenthal

algebra H3 (C , ±) for some Hurwitz algebra C . It remains to be shown that C C .

We consider the case where F has characteristic di¬erent from 2 (and leave the other

case as an exercise). For any bilinear form (x, y) ’ b(x, y) over L, let (TL/F )— (b)

be its transfer to F , i.e.,

(TL/F )— (b)(x, y) = TL/F b(x, y) .

The bilinear trace form of J(V, L) is the bilinear form:

T = (TL/F )— ( 1 L ) ⊥ (TL/F )— (bQ )

and bQ is extended from the bilinear form bQ0 = 2 L ⊥ δbNC over F (see Lemma

0

0

(??)). Let bNC = bC and bNC = bC . By Frobenius reciprocity (see Scharlau [?,

0

Theorem 5.6, p. 48]) we get

⊥ δb0 ).

T (TL/F )— ( 1 L ) ⊥ (TL/F )— ( 1 L ) — ( 2 L C

Since (TL/F )— ( 1 L ) = 1, 2, 2δ (see (??), (??)), it follows that

1, 2, 2δ ⊥ δb0 — 1, 2, 2δ ⊥ 2, 1, δ

T C

1, 2, 2 ⊥ bC — 2, δ, 2δ

1, 1, 1 ⊥ bC — 2, δ, 2δ .

Thus

bC — 2, δ, 2δ b C — ± 1 , ±2 , ±3 ,

since an isomorphism J(V, L) H3 (C , ±) implies that the corresponding trace

forms are isomorphic. The last claim then follows from Lemma (??) and Theo-

rem (??).

§39. The Tits Construction

Let K be a quadratic ´tale algebra with conjugation ι and let B be an associa-

e

tive separable algebra of degree 3 over K with a unitary involution „ (according to

an earlier convention, we also view K as a cubic separable K-algebra if char F = 3).

The generic norm NB of B de¬nes a cubic structure on B (as a K-algebra) and

restricts to a cubic structure on H(B, „ ). Let (u, ν) ∈ H(B, „ ) — K — be such that

NB (u) = ν„ (ν).

One can take for example (u, ν) = (1, 1). On the set

J(B, „, u, ν) = H(B, „ ) • B,

let 1 = (1, 0) and

N (a, b) = NB (a) + TK/F νNB (b) ’ TB abu„ (b)

(a, b)# = a# ’ bu„ (b), „ (ν)„ (b)# u’1 ’ ab

for (a, b) ∈ H(B, „ ) • B. Further let

T (a1 , b1 ), (a2 , b2 ) = TB (a1 a2 ) + TK/F TB b1 u„ (b2 ) .

§39. THE TITS CONSTRUCTION 523

(39.1) Theorem. The space J(B, „, u, ν) admits a Freudenthal cubic structure

with 1 as unit, N as norm, # as Freudenthal adjoint and T as bilinear trace form.

Furthermore we have SJ (a, b) = SB (a) ’ TB bu„ (b) for the quadratic trace SJ .

Reference: A characteristic-free proof is in Petersson-Racine [?, Theorem 3.4], see

also McCrimmon [?, Theorem 7]. The claim is also a consequence of Proposi-

tion (??) (see Corollary (??)) if char F = 3. The last claim follows from SJ (x) =

TJ (x# ) and TJ (x) = T (x, 1).

If char F = 2, the Jordan product of J(B, „, u, ν) is given by (see p. ??):

a1 b1

1

a2 2 (a1 a2 + a 2 a1 ) (a2 b1 )—

1

„ (u) „ (b1 ) — „ (b2 ) ν ’1

b2 (a1 b2 )— b1 ν„ (b2 ) + b2 ν„ (b1 ) + 2 —

where x = 1 TrdB (x) ’ x and x— denotes x as an element of the second com-

2

ponent B. The cubic structure J(B, „, u, ν) described in Theorem (??) is a Tits

construction or a Tits process and the pair (u, ν) is called an admissible pair for

(B, „ ). The following lemma describe some useful allowed changes for admissible

pairs.

(39.2) Lemma. (1) Let (u, v) be an admissible pair for (B, „ ). For any w ∈

B — , wu„ (w), νNB (w) is an admissible pair for (B, „ ) and (a, b) ’ (a, bw) is an

isomorphism

∼

J(B, „, u, ν) ’ J B, „, wu„ (w), νNB (w) .

’

(2) For any Tits construction J(B, „, u, ν), there is an isomorphic Tits construction

J(B, „, u , ν ) with NB (u ) = 1 = ν „ (ν ).

Proof : The ¬rst claim reduces to a tedious computation, which we leave as an

exercise (see also Theorem (??)). For the second, we take w = ν ’1 u in (??).

An exceptional Jordan algebra of dimension 27 of the form J(B, „, u, ν) where

B is a central simple algebra over a quadratic ¬eld extension K of F , is classically

called a second Tits construction. The case where K is not a ¬eld also has to be

considered. Let J(B, „, u, ν) be a Tits process with K = F — F , B = A — Aop where

A is either central simple or cubic ´tale over F and „ is the exchange involution.

e

By Lemma (??) we may assume that the admissible pair (u, ν) is of the form

1, (», »’1 ) , » ∈ F — . Projecting B onto the ¬rst factor A induces an isomorphism

of vector spaces

J(B, „, u, ν) A • A • A,

the norm is given by NJ (a, b, c) = NA (a) + »NA (b) + »’1 NA (c) ’ TA (abc) and the

Freudenthal adjoint on A • A • A reduces to

(a, b, c)# = (a# ’ bc, »’1 c# ’ ab, »b# ’ ca)

where a ’ a# is the Freudenthal adjoint of A+ (which is a cubic algebra!); thus

we have SJ (a, b, c) = SA (a) ’ TA (bc) and the bilinear trace form is given by

TJ (a1 , b1 , c1 ), (a2 , b2 , c2 ) = TA (a1 b1 ) + TA (a2 b3 ) + TA (a3 b2 ).

524 IX. CUBIC JORDAN ALGEBRAS

If char F = 2, the Jordan product is

(a1 , b1 , c1 ) q (a2 , b2 , c2 ) =

a1 q a2 + b1 c2 + b2 c1 , a1 b2 + a2 b1 + (2»)’1 (c1 — c2 ), c2 a1 + c1 a2 + 2 »(b1 — b2 ) ,

1

¯ ¯

where

a = 2 a — 1 = 1 TA (a) · 1 ’ 2 a.

¯1 1

2

Conversely we can associate to a pair (A, »), A central simple of degree 3 or cubic

´tale over F and » ∈ F — , a Freudenthal algebra J(A, ») = A • A • A, with norm,

e

Freudenthal product and trace as given above. The algebra J(A, ») is (classically)

a ¬rst Tits construction if A is central simple. Any ¬rst Tits construction J(A, »)

extends to a Tits process J A — Aop , σ, 1, (», »’1 ) over F — F . According to the

classical de¬nitions, we shall say that J(B, „, u, ν) is a second Tits process if K is

a ¬eld and that J(A, ») is a ¬rst Tits process.

(39.3) Remark. (See [?, p. 308].) Let (A, ») = A • A • A be a ¬rst Tits process.

To distinguish the three copies of A in J(A, »), we write

J(A, ») = A+ • A1 • A2

and denote a ∈ A as a, a1 , resp. a2 if we consider it as an element of A+ , A1 , or A2 .

The ¬rst copy admits the structure of an associative algebra, A1 (resp. A2 ) can be

characterized by the fact that it is a subspace of (A+ )⊥ (for the bilinear trace form)

such that a q a1 = ’a — a1 (resp. a2 q a = ’a — a2 ) de¬nes the structure of a left

A-module on A1 (resp. right A-module on A2 ).

(39.4) Proposition. For any second Tits process J(B, „, u, ν) over F , B a K-

algebra, J(B, „, u, ν) — K is isomorphic to the ¬rst Tits process J(B, ν) over K.

Conversely, any second Tits process J(B, „, u, ν) over F is the Galois descent of

the ¬rst Tits process (B, ν) over K under the ι-semilinear automorphism

(a, b, c) ’ „ (a), „ (c)u’1 , u„ (b) .

Proof : An isomorphism

∼

J(B, „, u, ν) — K ’ J(B, ν)

’

is induced by (a, b) ’ a, b, u„ (b) . The last claim follows by straightforward

computations.

(39.5) Examples. Assume that char F = 3.

(1) Any cubic ´tale F -algebra L can be viewed as a Tits construction over F ; if

e

√

3

L = F ( »), then L is isomorphic to the ¬rst Tits construction (F, »). In general

there exist a quadratic ´tale F -algebra K and some element ν ∈ K with NK (ν) = 1

e√

such that L — K K( 3 ν) and L is the second Tits construction (K, ι, 1, ν) (see

Proposition (??)).

(2) Let A be central simple of degree 3 over F . We write A as a crossed product

A = L • Lz • Lz 2 with L cyclic and z 3 = » ∈ F — , z = ρ( )z and ρ a generator of

Gal(L/F ); the map A ’ L • L • L given by a + bz + cz 2 ’ a, ρ(b), »ρ2 (c) is an

isomorphism of A+ with the ¬rst Tits construction (L, »).

§39. THE TITS CONSTRUCTION 525

(3) Let (B, „ ) be central simple of degree 3 with a distinguished unitary involution

„ over K. In view of Proposition (??) and Corollary (??), there exists a cubic ´tale

e

F -algebra L with discriminant ∆(L) K such that

B = L — K • (L — K)z • (L — K)z 2 z3 = » ∈ F —,

with „ (z) = z.

The K-algebra L — K is cyclic over K; let ρ ∈ Gal(L — K/K) be such that zξz ’1 =

ρ(ξ) for ξ ∈ L — K. We have

L1 = { ξ ∈ L — K | ρ —¦ (1 — ι)(ξ) = ξ } L,

L2 = { ξ ∈ L — K | ρ2 —¦ (1 — ι)(ξ) = ξ } L

and (1 — ι)(L1 ) = L2 , so that

H(B, „ ) = L • L1 • L2 L•L•L

and a check shows this is an isomorphism of H(B, „ ) with the ¬rst Tits construction

(L, »). Since the exchange involution on A — Aop is distinguished, we see that

H(B, „ ) is a ¬rst Tits construction if and only if „ is distinguished, if and only if

SB |H(B,„ )0 has Witt index at least 3 (see Proposition (??) for the last equivalence).

(4) Let (B, „ ) be central simple with a unitary involution over K and assume that

H(B, „ ) contains a cyclic ´tale algebra L over F . By Albert [?, Theorem 1] we may

e

write B as a crossed product

B = L — K • (L — K)z • (L — K)z 2

(39.6)

with z 3 = ν ∈ K — such that NK (ν) = 1; furthermore the involution „ is determined

by „ (z) = uz ’1 with u ∈ L such that NB (u) = 1. In this case H(B, „ ) is isomorphic

to the second Tits process (L — K, 1 — ιK , u, ν).

(5) A Tits construction J = J(L — K, 1 — ιK , u, ν) with L cubic ´tale is of di-

e

mension 9, hence by Theorem (??) it is of the form H(B, „ ) for a central simple

algebra B of degree 3 over an ´tale quadratic F -algebra K1 and a unitary involution

e

„ . We may describe (B, „ ) more explicitly: if L is cyclic, K1 = K, and B is as

in (??). If L is not cyclic, we replace L by L2 = L — ∆(L), where ∆(L) is the

discriminant of L, and obtain (B2 , „2 ) over ∆(L) from (??). Let φ be the descent

φ

on B2 given by φ = 1 — ι∆(L) — ιK on L — ∆(L) — K and φ(z) = z ’1 . Then B = B2

(A — Aop , exchange) if and

and K1 = ∆(L) — K. In particular we have (B, „ )

only if ∆(L) K.

(6) Let J be a Freudenthal algebra of dimension 9 over F and let L be a cubic ´tale

e

subalgebra of J. We may describe J as a Tits construction J(L — K, 1 — ιK , u, ν)

as follows. Let J = L • V be the Springer decomposition induced by L. Then V is

a twisted composition (V, L, Q, β) and V is of dimension 2 over L; by Proposition

(??) V admits the structure of a hermitian L — K-space for some quadratic ´tale e

—

F -algebra K. Let V = (L — K)v; let u = Q(v) ∈ L and let β(v) = xv, x ∈ L — K.

It follows from bQ v, β(v) ∈ F that (x + x)u ∈ F , where x ’ x is the extension of

the conjugation ιK of K to L — K. Similarly Q β(v) = u# implies that xxu2 =

NL/F (u) ∈ F — . Both imply that xu (or xu) lies in K and NK/F (xu) = nL/F (u).

Let J be the Tits construction J(L — K, 1 — ιK , u, xu). The map J ’ J given by

(a, b) ’ (a, bv) is an isomorphism of Jordan algebras.

(7) A ¬rst Tits construction J(A, 1) with A cubic ´tale or central simple of de-

e

gree 3 is always a split Freudenthal algebra: this is clear for cubic ´tale algebras

e

by Example (??). So let A be central simple. Taking a ∈ A such that a3 ∈ F — ,

526 IX. CUBIC JORDAN ALGEBRAS

we see that a# = a2 , so that (a, a, a)# = 0 and, by Exercise ?? of this chapter

J(A, 1) is reduced. Theorem (??) then implies that J(A, 1) H3 (C, ±). Let L1

be a cubic extension which splits A. By Theorem (??) C — L1 is split, hence by

Springer™s theorem for quadratic forms, C is split. The claim then follows from

Corollary (??).

39.A. Symmetric compositions and Tits constructions. In this section,

we assume that char F = 3. The aim is to show that Tits constructions with admis-

sible pairs (1, ν) are also Springer constructions. We start with a ¬rst Tits construc-

tion (A, »); let L = F [X]/(X 3 ’ ») be the cubic Kummer extension associated with

» ∈ F — , set, as usual, A0 = { x ∈ A | TA (x) = 0 } and let “(A, ») = (A0 —L, L, N, β)

be the twisted composition of type 1A2 induced by A and » (see (??)). Let

J “(A, ») = L • “(A, ») = L • A0 — L = L — A

be the Freudenthal algebra obtained from “(A, ») by the Springer construction. If

v is the class of X modulo (X 3 ’ ») in L, (1, v, v ’1 ) is a basis of L as vector space

over F and we write elements of L — A as linear combinations a + v — b + v ’1 — c,

with a, b, c ∈ A.

∼

(39.7) Proposition. The isomorphism φ : J “(A, ») ’ J(A, ») = A • A • A

’

given by

a + v — b + v ’1 — c ’ (a, b, c) for a, b, c ∈ A

is an isomorphism of Freudenthal algebras.

Proof : We use the map φ to identify L as an ´tale subalgebra of J(A, ») and get a

e

corresponding Springer decomposition

J(A, ») = L • L — A0 .

It follows from Theorem (??) and from the description of a twisted composition

“(A, ») of type 1A2 given in § ?? that φ restricts to an isomorphism of twisted

∼

compositions “(A, ») ’ L — A0 , hence the claim.

’

(39.8) Corollary. Tits constructions are Freudenthal algebras.

Proof : We assume char F = 3. By descent we are reduced to ¬rst Tits construc-

tions, hence the claim follows from Proposition (??).

(39.9) Corollary. Let G be a split simple group scheme of type F4 . Jordan alge-

bras which are ¬rst Tits constructions are classi¬ed by the image of the pointed

set H 1 (F, PGL3 —µ3 ) in H 1 (F, G) under the map PGL3 —µ3 ’ G induced by

(A, ») ’ J(A, »).

Not all exceptional Jordan algebras are ¬rst Tits construction (see Petersson-

Racine [?] or Proposition (??)). Thus the cohomology map in (??) is in general

not surjective (see also Proposition (??)).

We now show that the Springer construction associated with a twisted composi-

tion of type 2A2 is always a second Tits construction. Let (B, „ ) be a central simple

algebra with a unitary involution over a quadratic ´tale F -algebra K. Let ν ∈ K

e

be such that NK (ν) = 1; let L be as in Proposition (??), (??), and let “(B, „, ν)

be the corresponding twisted composition, as given in Proposition (??), (??).

∼

(39.10) Proposition. There exists an isomorphism L•“(B, „, ν) ’ J(B, „, 1, ν).

’

§39. THE TITS CONSTRUCTION 527

Proof : The twisted composition “(B, „, ν) over F is de¬ned by descent from the

twisted composition “(B, ν) over K (see Proposition (??)); similarly J(B, „, 1, ν) is

de¬ned by descent from J(B, ν) (see Proposition (??)). The descents are compatible

with the isomorphism given in Proposition (??), hence the claim.

39.B. Automorphisms of Tits constructions. If J is a vector space over F

with some algebraic structure and A is a substructure of J, we write Aut F (J, A)

for the group of F -automorphisms of J which maps A to A and by Aut F (J/A) the

group of automorphisms of J which restrict to the identity on A. The corresponding

group schemes are denoted Aut(J, A) and Aut(J/A).

(39.11) Proposition (Ferrar-Petersson, [?]). Let A be central simple of degree 3

and let J0 = J(A, »0 ) be a ¬rst Tits construction. The sequence of group schemes

γ ρ

1 ’ SL1 (A) ’ Aut(J0 , A+ ) ’ Aut(A+ ) ’ 1

’ ’

where γ(u)(a, a1 , a2 ) = (a, a1 u’1 , ua2 ) for u ∈ SL1 (A)(R), R ∈ Alg F , and ρ is the

restriction map, is exact.

Proof : Let R ∈ Alg F ; exactness on the left (over R) and ρR —¦ γR = 1 is obvious.

Let J0 = A+ • A1 • A2 and let · be an automorphism of J0 R which restricts to the

identity on A+ . It follows from Remark (??) that · stabilizes A1 R and A2 R , so there

R

exist linear bijections ·i : Ai R ’ Ai R such that ·(a, a1 , a2 ) = a, ·1 (a1 ), ·2 (a2 ) .

Expanding · a — (0, a1 , a2 ) in two di¬erent ways shows that

·1 (aa1 ) = a·1 (a1 ) and ·2 (a2 a) = ·2 (a2 )a.

Hence there are u, v ∈ A— such that ·1 (a1 ) = a1 v and ·2 (a2 ) = ua2 . Comparing the

R

¬rst components of · (0, 1, 1)# = ·(0, 1, 1) # yields v = u’1 . Since · preserves

the norm we have u ∈ SL1 (A)(R). To conclude, since Aut(A+ ) is smooth (see

the comments after the exact sequence (??)), it su¬ces to check by (??) that ρalg

is surjective. In fact ρ is already surjective: let φ ∈ Aut(A+ ), hence, by the

exact sequence (??), φ is either an automorphism or an anti-automorphism of A.

In the ¬rst case, φ(a, a1 , a2 ) = φ(a), φ(a1 ), φ(a2 ) extends φ to an element of

Aut(J0 , A+ ). In the second case, A is split, so some u ∈ A— has NA (u) ∈ F —2 and

φ(a, a1 , a2 ) = φ(a), φ(a2 )u’1 , uφ(a1 ) extends φ.

√ √

Now, let L = F ( 3 »). We embed L = F (v), v = 3 », in J(A, ») = A • A • A

through v ’ (0, 1, 0) and v ’1 ’ (0, 0, 1). Furthermore we set

(A— — A— )Det = { (f, g) ∈ A— — A— | NrdA (f ) = NrdA (g) }.

(39.12) Corollary. (1) We have AutF J(A, »), A+ = (A— — A— )Det /F — , where

F — operates diagonally, if A is a division algebra and

Det

AutF J(A, »), A+ = GL3 (F ) — GL3 (F ) /F — Z/2Z

if A = M3 (F ). The action of Z/2Z on a pair (f, g) is given by

(f, g) ’ (f ’1 )t , (g ’1 )t .

The action of (f, g) on J(A, ») is given by

(f, g)(a, b, c) = (f af ’1 , f bg ’1 , gcf ’1 )

and the action of Z/2Z by „ (a, b, c) = (at , ct , bt ).

528 IX. CUBIC JORDAN ALGEBRAS

(2) We have AutF J(A, »), A+ , L AutF (A)/F — — µ3 if A is a division algebra

and

AutF J(A, »), A+ , L Z/2Z

PGL3 (F ) — µ3

where the action of Z/2Z is given by „ (f, µ) = ([f t ]’1 , µ’1 ) if A = M3 (F ).

Proof : (??) If φ ∈ AutF J(A, »), A+ restricts to an automorphism φ of A, we

write φ = Int(f ) and (??) follows from Proposition (??). If φ restricts to an

anti-automorphism φ of A, we replace φ by φ —¦ „ and apply the preceding case.

(??) We assume that A = M3 (F ). By (??) we can write any element φ of

AutF J(A, »), A+ , L as [f, g] with f , g ∈ GL3 (F ). Since φ restricts to an auto-

morphism of L, we must have φ(u) = ρu±1 , ρ ∈ F — . Since „ (u) = u’1 , we may

assume that φ(u) = ρu (replace φ by φ„ ). It follows that ρ3 = 1 and ρ ∈ µ3 (F ).

Since φ (0, 1, 0) = (0, f g ’1 , 0) = (0, ρ’1 , 0) we get g = ρf with ρ ∈ µ3 (F ). The

map (f, ρ) ’ (f, ρf ) then induces the desired isomorphism.

(39.13) Remark. If F contains a primitive cubic root of unity, we may identify

Z/2Z with PGL3 (F ) S3

µ3 with A3 (as Galois-modules) and PGL3 (F ) — µ3

where S3 operates through its projection on Z/2Z. In particular we get for the split

Jordan algebra Js

AutF Js , M3 (F )+ , F — F — F = PGL3 (F ) S3 .

On the other hand we have

AutF (Js , F — F — F ) = Spin8 (F ) S3

(see Corollary (??)), so that

Det

/F — Z/2Z © Spin8 (F )

PGL3 (F ) S3 = GL3 (F ) — GL3 (F ) S3 ‚ G(F )

where G = Aut(Js ) is a simple split group scheme of type F4 .

(39.14) Theorem. (1) (Ferrar-Petersson) Let J0 = J(A, »0 ) be a ¬rst Tits con-

struction with A a central simple associative algebra of degree 3. The cohomology

set H 1 F, Aut(J0 , A+ ) classi¬es pairs (J , I ) with J an Albert algebra over F

and I is a central simple Jordan algebra of dimension 9 over F . The cohomology

set H 1 F, Aut(A+ ) classi¬es central simple Jordan algebra of dimension 9 over F .

The sequence of pointed sets

ρ1

ψ

1 ’ F — /NA (A— ) ’ H 1 F, Aut(J0 , A+ ) ’ H 1 F, Aut(A+ )

’ ’

is exact and ψ([»]) = [J(A, »»0 ), A+ ], ρ1 ([J , A ]) = [A ].

(2) Let J be an Albert algebra containing a subalgebra A+ for A central simple of

∼

degree 3. There exist » ∈ F — and an isomorphism φ : J ’ J(A, ») which restricts

’

to the identity on A+ .

(3) J(A, ») is a division algebra if and only if » is not the reduced norm of an

element from A.

Proof : We follow Ferrar-Petersson [?]. (??) We assume for simplicity that F is a

¬eld of characteristic not 2, so that J0 is an F -algebra with a multiplication m.

Let F be the ¬‚ag A+ ‚ J0 and let W = HomF (J0 — J0 , J0 ). We let G = AutF (F)

act on F • W through the natural action. Let w = (0, m). Since AutG (w) =

+

Aut(J0 , A+ ) and since (J0 , A+ )Fsep (J , I )Fsep ( (Js , M3 )Fsep ), the ¬rst claim

follows from (??) and from Corollary (??). The fact that H 1 F, Aut(A+ ) classi¬es

central simple Jordan algebra of dimension 9 over F then is clear.

§39. THE TITS CONSTRUCTION 529

The exact sequence is the cohomology exact sequence associated with the se-

quence (??), where the identi¬cation (??) of F — /NA (A— ) with H 1 F, SL1 (A)

is as follows: let » ∈ F — and let v ∈ A— be such that NAsep (v) = ». Then

sep

± : Gal(Fsep /F ) ’ SL1 (A)(Fsep ) such that ±(g) = v ’1 g(v) is the cocycle induced

by ». The image of the class of » ∈ F — in H 1 F, Aut(J0 , A+ ) is the class of the

cocycle β given by β(g)(a, a1 , a2 ) = a, a1 g(v ’1 )v, v ’1 g(v)a2 . Let γ ∈ GL (J0 )sep

be given by

γ(a, a1 , a2 ) = (a, a1 v ’1 , va2 ),

then β(g) = g(γ ’1 )γ, and, setting J = J(A, »»0 ), one can check that

∼

γ : (J0 , A+ )Fsep ’ (J, A+ )Fsep

’

is an isomorphism, hence (J, A+ ) is the F -form of (J0 , A+ ) given by the image of ».

(??) We set »0 = 1 in (??). Let J be a reduced Freudenthal algebra. By

Theorem (??), we have (J, A+ )Fsep (J0 , A+ )Fsep . Therefore (J, A+ ) is a form of

(J0 , A+ ) and its class in H 1 F, Aut(J0 , A+ ) belongs to the kernel of ρ1 , hence

in the image of ψ. Thus by (??) there exists » ∈ F — such that J(A, »), A+

(J, A+ ), as claimed.

(??), similarly, follows from (??), since J(A, 1) is split (see Example (??),

(??)).

(39.15) Remark. By (??.??), J(A, ») is a division algebra if and only if A is a

division algebra and » is not a reduced norm of A. Examples can be given over

a purely transcendental extension of degree 1: Let F0 be a ¬eld which admits a

division algebra A0 of degree 3 and let A = A0 — F0 (t). Then the Albert algebra

J(A, t) is a division algebra (see Jacobson, [?, p. 417]).

The analogue of Proposition (??) for second Tits constructions is

(39.16) Proposition. Let J0 = J(B, „, u0 , ν0 ) be a second Tits construction and

let „ = Int(u0 ) —¦ „ . The sequence

γ ρ

1 ’ SU(B, „ ) ’ Aut J0 , H(B, „ ) ’ Aut H(B, „ ) ’ 1,

’ ’

where γR (u)(a, b)R = (a, bu’1 )R and ρR is restriction, is exact.

Proof : (??) follows from (??) by descent, using Proposition (??).

To get a result corresponding to Theorem (??) for second Tits constructions, we

recall that the pointed set H 1 F, SU(B, „ ) classi¬es pairs (u, ν) ∈ Sym(B, „ )— —

K — with NB (u) = NK (ν) under the equivalence ≈, where

(39.17)

(u, ν) ≈ (u , ν ) if and only if u = bu„ (b) and ν = ν · NB (b) for some b ∈ B —

(see (??)). As in (??) we set

SSym(B, „ )— = { (u, ν) ∈ H(B, „ ) — K — | NB (u) = NK (ν) }.

(39.18) Theorem. Let J0 = J(B, „, u0 , ν0 ).

(1) The sequence of pointed sets

ψ

1 ’ SSym(B, „ )— /≈ ’

’

ρ1

H 1 F, Aut J0 , H(B, „ ) ’ H 1 F, Aut H(B, „ ) ,

’

530 IX. CUBIC JORDAN ALGEBRAS

where ψ([u, ν]) = [J(B, „, uu0 , νν0 ), H(B, „ )] and ρ1 ([J , A ]) = [A ], is exact.

(2) Let J be a Freudenthal algebra of dimension 27 containing a subalgebra H(B, „ )

for (B, „ ) central simple of degree 3 with a unitary involution. There exist an