admissible pair (u, ν) ∈ SSym(B, „ )— and an isomorphism φ : J ’ J(B, „, u, ν)

’

which restricts to the identity on H(B, „ ).

(3) J(B, „, u, ν) is a division algebra if and only if u is not the reduced norm of an

element from B — .

Proof : The proof of (??) is similar to the one of Theorem (??) and we skip it.

Any Hurwitz algebra can be obtained by successive applications of the Cayley-

Dickson process, starting with F . The next result, which is a special case of a

theorem of Petersson-Racine [?, Theorem 3.1], shows that a similar result holds

for Freudenthal algebras of dimension 3, 9 and 27 if Cayley-Dickson processes are

replaced by Tits processes:

(39.19) Theorem (Petersson-Racine). Assume that char F = 3. Any Freudenthal

algebra of dimension 3, 9 or 27 can be obtained by successive applications of the

Tits process. In particular any exceptional Jordan algebra of dimension 27 is of the

form H(B, „ ) • B where B is a central simple of degree 3 over a quadratic ´tale

e

F -algebra K with a unitary involution.

Proof : A Freudenthal algebra of dimension 3 is a cubic ´tale algebra, hence the

e

claim follows from Example (??), (??), if dim J = 3. The case dim J = 9 is

covered by Example (??), (??). If J has dimension 27 and J contains a Freudenthal

subalgebra of dimension 9 of the type H(B, „ ), then by Theorem (??), there exists

a pair (u, ν) such that J (B, „, u, ν). Thus we are reduced to showing that J

contains some H(B, „ ). If J is reduced this is clear, hence we may assume that J

is not reduced. Then (see the proof of (??)) F is an in¬nite ¬eld. Let L be a cubic

´tale F -subalgebra of J and let J = L • V , V = (V, L, Q, β), be the corresponding

e

Springer decomposition. For some v ∈ V , the set {v, β(v)} is linearly independent

over L since Q v, β(v) is anisotropic and by a density argument (F is in¬nite) we

may also assume that Q restricted to U = Lv • Lβ(v) is L-nonsingular. Thus J

contains a Springer construction J1 = J(L, U ) of dimension 9. In view of the 9-

dimensional case J1 is a Tits construction and by Example (??), (??), J1 H(B, „ )

for some central simple algebra (B, „ ) of degree 3 with unitary involution, hence

the claim.

Jordan algebras of the form L+ (L cubic ´tale of dimension 3), A+ (A central

e

simple of degree 3), or H(B, „ ) (B central simple of degree 3 with an involution

of the second type) are “generic subalgebras ” of Albert algebras in the following

sense:

(39.20) Proposition. Let J be an Albert algebra.

(1) There is a Zariski-open subset U of J such that the subalgebra generated by x

is ´tale for all x ∈ U .

e

(2) There is a Zariski-open subset U of J such that the subalgebra generated by

x ∈ U and y ∈ U is of the form A+ , for A central simple of degree 3 over F , or of

the form H(B, „ ) for B central simple over a quadratic separable ¬eld extension K

and „ a unitary involution.

Proof : The ¬rst claim is already in (??), (??). The second follows from the proof

of (??).

§40. COHOMOLOGICAL INVARIANTS 531

(39.21) Remark. The element v in the proof of (??) is such that v and β(v) are

linearly independent over L (see Proposition (??)). If v is such that β(v) = »v for

» in L, then J is reduced by Theorem (??) and (L, v) generates a 6-dimensional

subalgebra of J of the form H3 (F, ±) (Soda [?, Theorem 2]). Such an algebra is

not “generic”.

(39.22) Remark. If char F = 3, the only Freudenthal algebras which cannot be

obtained by iterated Tits constructions are separable ¬eld extensions of degree 3

(see [?, Theorem 3.1]). We note that Petersson and Racine consider the more

general case of simple cubic Jordan structures (not just Freudenthal algebras) in

[?, Theorem 3.1].

The Albert algebra Js = J M3 (F ), 1 is split, thus G = Aut J M3 (F ), 1 is

a simple split group scheme of type F4 .

(39.23) Proposition. The pointed set H 1 F, (GL3 — GL3 )Det / Gm Z/2Z clas-

si¬es pairs (J, H B, „ ) where J is an Albert algebra, B ‚ J is central simple with

unitary involution „ over a quadratic ´tale algebra K. The map

e

H 1 F, (GL3 — GL3 )Det / Gm Z/2Z ’ H 1 (F, G),

induced by AutF J M3 (F ), 1 , M3 (F )+ ’ AutF J M3 (F ), 1 and which asso-

ciates the class of J to the class of J, H(B, „ ) is surjective.

Proof : The ¬rst claim follows from Corollary (??) and Theorem (??), the second

then is a consequence of Theorem (??).

(39.24) Remark. Let J be an Albert algebra. We know that J J(B, „, u, β)

for some datum (B, „, u, ν). The datum can be reconstructed cohomologically as

follows. Let [±] ∈ H 1 F, (GL3 — GL3 )Det / Gm Z/2Z be a class mapping to [J].

The image [γ] ∈ H 1 (F, Z/2Z) of [±] under the map in cohomology induced by

the projection (GL3 — GL3 )Det / Gm Z/2Z ’ Z/2Z de¬nes the quadratic exten-

sion K. Pairs (J, H B, „ ) with ¬xed K are classi¬ed by

H 1 F, (GL3 — GL3 )Det / Gm γ

and the projection on the ¬rst factor gives an element of H 1 F, (PGL3 )γ , hence

by Remark (??) a central simple K-algebra B with unitary involution „ . We ¬nally

get (u, ν) from the exact sequence (??.??).

§40. Cohomological Invariants

In this section we assume that F is a ¬eld of characteristic not 2. Let J =

H3 (C, ±) be a reduced Freudenthal algebra of dimension > 3. Its bilinear trace

form is given by

T = 1, 1, 1 ⊥ bC — ’b, ’c, bc

where bC is the polar of NC . As known from Corollary (??) and Theorem (??),

the P¬ster forms bC and bC — b, c determine the isomorphism class of J. Let

dimF C = 2i , let fi (J) ∈ H i (F, Z/2Z) be the cohomological invariant of the

P¬ster form bC and let fi+2 (J) ∈ H i+2 (F, Z/2Z) be the cohomological invariant

of bC — b, c . These two invariants determine J up to isomorphism. Observe

that Freudenthal algebras of dimension 3 with zero divisors are also classi¬ed by a

cohomological invariant: Such an algebra is of the form F + — K + and is classi¬ed

532 IX. CUBIC JORDAN ALGEBRAS

by the class of K f1 (K) ∈ H 1 (F, S2 ). We now de¬ne the invariants f3 (J) and

f5 (J) for division algebras J of dimension 27 (and refer to Proposition (??), resp.

Theorem (??) for the corresponding invariants of algebras of dimension 3, resp. 9).

We ¬rst compute the bilinear trace form of J.

(40.1) Lemma. (1) Let (B, „ ) be a central simple algebra of degree 3 over a quad-

ratic ´tale F -algebra K with a unitary involution and let T„ be the bilinear trace

e

form of the Jordan algebra H(B, „ ). Then

for b, c ∈ F —

T„ 1, 1, 1 ⊥ bK/F — ’b, ’c, bc

where bK/F stands for the polar of the norm of K.

(2) Let J be a Freudenthal algebra of dimension 27 and let T be the bilinear trace

form of J. There exist a, b, c, e, f ∈ F — such that

T 1, 1, 1 ⊥ 2 — a, e, f — ’b, ’c, bc .

Proof : (??) follows from Proposition (??).

(??) By Theorem (??) we may assume that J is a second Tits construction

J(B, „, u, µ), so that

T (x, y), (x , y ) = T„ (x, x ) + TK/F TrdB yu„ (y )

for x, x ∈ H(B, „ ) and y, y ∈ B. By Lemma (??), (??), we may assume that

NrdB (u) = 1. Let „ = Int(u’1 ) —¦ „ . By (??) the trace form of H(B, „ ) is of the

form

T„ = 1, 1, 1 ⊥ bK/F — ’e, ’f, ef

for e, f ∈ F — . Let T„,„ (x, y) = TK/F TrdB xu„ (y) for x, y ∈ B. We claim that

T„,„ bK/F — ’b, ’c, bc — ’e, ’f, ef .

The involution „ is an isometry of the bilinear form T„,„ with the bilinear form

(TK/F )— (TB,„,u ) where TB,„,u (x, y) = TrdB „ (x)uy . Thus it su¬ces to have an

isomorphism of hermitian forms

TB,„,u ’b, ’c, bc —K ’e, ’f, ef

K K

since

(TK/F )— ±1 , . . . ±n = bK/F — ±1 , . . . ±n .

K

In view of Proposition (??) the unitary involution „ — „ on B —K ι B corre-

sponds to the adjoint involution on EndK (B) of the hermitian form T(B,„,u) under

the isomorphism „— : B —K ι B ’ EndK (B). By the Bayer-Lenstra extension (??)

of Springer™s theorem, we may now assume that B = M3 (K) is split, so that by

Example (??) „ is the adjoint involution of ’b, ’c, bc K and „ is the adjoint invo-

lution of ’e, ’f, ef K . This shows that TB,„,u and ’b, ’c, bc K —K ’e, ’f, ef K

are similar hermitian forms and it su¬ces to show that they have the same deter-

minant. By Corollary (??) the form T(B,„,u) has determinant the class of Nrd(u),

which, by the choice of u, is 1. Thus we get

T T„ ⊥ T„,„ 1, 1, 1 ⊥ 2 — a, e, f — ’b, ’c, bc

√

where K = F ( a).

§40. COHOMOLOGICAL INVARIANTS 533

(40.2) Theorem. (1) Let F be a ¬eld of characteristic not 2. For any Freudenthal

algebra J of dimension 3 + 3 · 2i , 1 ¤ i ¤ 3, there exist cohomological invariants

fi (J) ∈ H i (F, Z/2Z) and fi+2 (J) ∈ H i+2 (F, Z/2Z) which coincide with the invari-

ants de¬ned above if J is reduced.

(2) If J = J(B, „, u, ν) is a second Tits construction of dimension 27, then f3 (J)

is the f3 -invariant of the involution „ = Int(u) —¦ „ of B.

√

Proof : (??) Let K = F ( a). With the notations of Lemma (??), the invariants are

given by the cohomological invariants of the P¬ster forms a , resp. a, b, c if J

has dimension 9 and the cohomological invariants of a, e, f , resp. a, e, f — b, c

if J has dimension 27. The fact that these are fi -, resp. fi+2 -invariants of J follows

as in Corollary (??).

Claim (??) follows from the computation of T„ .

(40.3) Corollary. If two second Tits constructions J(B, „, u1 , ν1 ), J(B, „, u2 , ν2 )

of dimension 27 corresponding to di¬erent admissible pairs (u1 , ν1 ), (u2 , ν2 ) are

isomorphic, then there exist w ∈ B — and » ∈ F — such that »u2 = wu1 „ (w). If

furthermore Nrd(u1 ) = Nrd(u2 ), then w can be chosen such that u2 = wu1 „ (w).

Proof : Let „i = Int(ui ) —¦ „ , i = 1, 2. In view of Theorem (??), (??), and

Theorem (??), the involutions „1 and „2 of B are isomorphic, hence the ¬rst

claim. Taking reduced norms on both sides of »u2 = wu1 „ (w) we get »3 =

’1

NrdB (w) NrdB „ (w) and » is of the form » » . Replacing w by w» , we get

the second claim.

(40.4) Remark. Corollary (??) is due to Parimala, Sridharan and Thakur [?].

As we shall see in Theorem (??) (which is due to the same authors) w can in fact

be chosen such that u2 = wu1 „ (w) and ν2 = ν1 NrdB (w) so that the converse of

Lemma (??) holds.

For an Albert algebra J with invariants f3 (J) and f5 (J), the condition f3 (J) =

0 obviously implies f5 (J) = 0. More interesting are the following two propositions:

(40.5) Proposition. Let J be an Albert algebra. The following conditions are

equivalent:

(1) J is a ¬rst Tits construction, J = J(A, »).

(2) There exists a cubic extension L/F such that JL splits over L.

(3) The Witt index w(T ) of the bilinear trace form T of J is at least 12.

(4) f3 (J) = 0.

(5) For any Springer decomposition J = J(V, L) with corresponding twisted com-

position “ = (V, L, Q, β), we have wL (Q) ≥ 3.

Proof : (??) ’ (??) Choose L which splits A.

(??) ’ (??) By Springer™s Theorem we may assume that J is split. The claim

then follows from the explicit computation of the bilinear trace form given in (??).

(??) ’ (??) We have

T 1, 1, 1 ⊥ a, e, f — ’b, ’c, bc .

Thus, if w(T ) ≥ 12, the anisotropic part ban of a, e, f — ’b, ’c, bc has at most

dimension 6; since ban ∈ I 3 F , the theorem of Arason-P¬ster (see Lam [?, p. 289])

shows that ban = 0 in W F . Lemma (??) then implies that a, e, f is hyperbolic,

hence f3 (J) = 0.

534 IX. CUBIC JORDAN ALGEBRAS

(??) ’ (??) Let J = J(V, L) be a Springer decomposition of J for a twisted

composition “ = (V, L, Q, β). To check (??), we may assume that L is not a ¬eld:

if L is a ¬eld we may replace L by L — L. Then (V, Q) is similar to a Cayley

composition (V0 , Q0 ) with

Q0 = 1 ⊥ δ — (C, n)0

(see Theorem (??) and Lemma (??)). Since f3 (J) is the cohomological invariant of

the norm of C, we get (??).

(??) ’ (??) We may assume that J is a division algebra. (??) also implies

(??) and a reduced algebra with f3 = 0 is split. Let x ∈ V with Q(x) = 0. We

have Q β(x) = 0 and Q x, β(x) = 0 (by Proposition (??) and Theorem (??),

since J is a division algebra). Thus U = Lx • Lβ(x) is a 2-dimensional twisted

composition. By Proposition (??), Q|U is the trace of a hermitian 1-form over L—K

for some quadratic ´tale F -algebra K. Furthermore K is split if Q|U is isotropic.

e

Now the Springer construction J1 = L • U is a 9-dimensional Freudenthal algebra

and the Witt index of the bilinear trace form of J1 is at least 2. As shown in

Example (??), (??), J1 is a second Tits construction J1 = J(L — K, 1 — ιK , u1 , ν1 )

and by Example (??), (??), J1 H(B1 , „1 ) where B1 is central simple of degree 3

with a unitary involution „1 . Moreover the center K1 of B1 is the discriminant

algebra ∆(L) (since K as above is split). Since the trace on H(B1 , „1 ) is of Witt

index ≥ 2, „1 is distinguished (Proposition (??)). Furthermore, by Theorem (??),

J is a second Tits construction J = J(B1 , „1 , u, ν) for the given (B1 , „1 ). We

have f3 Int(u) —¦ „1 = f3 (J) and since (??) implies (??), „ = Int(u) —¦ „1 is also

„ and there exist » ∈ F — and w ∈ B — such

distinguished. By Theorem (??) „

that u = »w„ (w). By Lemma (??) we may assume that u = » ∈ F — . Then the

Tits construction J2 = J(L—K1 , 1—ιK1 , », ν) is a subalgebra of J = J(B1 , „1 , u, ν).

By Example (??) of (??), J2 H(B2 , „2 ) and the center of B2 is K1 — ∆(L). Since

K1 ∆(L), J2 (A — Aop , exchange) and we conclude using Theorem (??).

(40.6) Remark. The equivalence of (??) and (??) in (??) is due to Petersson-

Racine [?, Theorem 4.7] if F contains a primitive cube root of unity. The trace

form then has maximal Witt index.

(40.7) Proposition. Let J be an Albert algebra. The following conditions are

equivalent:

(1) J = J(B, „, u, ν) is a second Tits construction with „ a distinguished unitary

involution of B.

(2) The Witt index w(T ) of the bilinear trace form T of J is at least 8.

(3) f5 (J) = 0.

Proof : We use the notations of the proof of Lemma (??).

(??) ’ (??) The bilinear form bK/F — b, c is hyperbolic if „ is distinguished.

Thus T„,„ has Witt index at least 6. By Proposition (??), T„ has Witt index at

least 2, hence the claim.

(??) ’ (??) If w(T ) ≥ 8, a, e, f is isotropic, hence f5 (J) = 0.

The proof of (??) ’ (??) goes along the same lines.

(??) ’ (??) We assume that J is a division algebra. Let J = L • V be a

Springer decomposition of J; since (??) ’ (??) holds, we get that T |V is isotropic.

We may choose x ∈ V such that T (x, x) = 0 and such that U = Lx • β(x) is a 2-

dimensional twisted composition. Then J1 = L•U is a Freudenthal subalgebra of J

§40. COHOMOLOGICAL INVARIANTS 535

of dimension 9, hence of the form H(B, „ ). Since w(T |J1 ) ≥ 2, „ is distinguished.

The claim then follows from Theorem (??).

We now indicate how one can associate a third cohomological invariant g 3 (J)

to an Albert algebra J. We refer to Rost [?], for more information (see also the Reference

paper [?] of Petersson and Racine for an elementary approach). By Theorem (??), missing: connect

we may assume that J = J(B, „, u, ν) is a second Tits process and by Lemma (??) to H3

√

that NrdB (u) = νι(ν) = 1. Let Lν be the descent of K( 3 ν) under the action given

√

by ιK on K and ξ ’ ξ ’1 for ξ = 3 ν. Then Lν de¬nes a class in H 1 (F, µ3[K] )

by Proposition (??). On the other hand, the algebra with involution (B, „ ) de-

termines a class g2 (B, „ ) ∈ H 2 (F, µ3[K] ) by Proposition(??). Since there exists a

canonical isomorphism of Galois modules µ3[K] — µ3[K] = Z/3Z (with trivial Galois

action on Z/3Z), the cup product g2 (B, „ ) ∪ [ν] de¬nes a cohomology class g3 in

H 3 (F, Z/3Z). If K = F — F , B = A — Aop and ν = (», »)’1 , then [A] ∈ H 2 (F, µ3 ),

[»] ∈ H 1 (F, µ3 ) and we have g3 = [A] ∪ [»] ∈ H 3 (F, Z/3Z). The following result is

due to Rost [?]:

(40.8) Theorem. (1) The cohomology class g3 ∈ H 3 (F, Z/3Z) is an invariant of

the Albert algebra J = J(B, „, u, ν), denoted g3 (J).

(2) We have g3 (J) = 0 if and only if J has zero divisors.

(40.9) Remark. By de¬nition we have g3 = g2 (B, „ )∪[ν] and by Proposition (??)

we know that g2 (B, „ ) = ± ∪ β with ± ∈ H 1 (F, Z/3Z[K] ) and β ∈ H 1 (F, µ3 ); thus

g3 ∈ H 1 (F, Z/3Z[K] ) ∪ H 1 (F, µ3 ) ∪ H 1 (F, µ3[K] )

is a decomposable class.

It is conjectured that the three invariants f3 (J), f5 (J) and g3 (J) classify

Freudenthal algebras of dimension 27. This is the case if g3 = 0; then J is re-

duced, J H3 (C, ±), f3 (J) = f3 (C) determines C, f3 (J), f5 (J) determine the trace

and the claim follows from Theorem (??).

Theorem (??) allows to prove another part of the converse to Lemma (??)

which is due to Petersson-Racine [?, p. 204]:

(40.10) Proposition. If J(B, „, u, ν) J(B, „, u , ν ) then ν = ν Nrd(w) for

some w ∈ B — .

Proof : The claim is clear if B is not a division algebra, since then the reduced

norm map is surjective. Assume now that J = J(B, „, u, ν) J = J(B, „, u , ν ).

By (??), (??), we may assume that NK/F (ν) = 1 = NK/F (ν ). Let L, resp. L , be

the cubic extensions of F determined by ν, resp. ν , as in Proposition (??). We have

[B] ∪ [L] = g3 (J) = g3 (J ) = [B] ∪ [L ], hence [B] ∪ ([L ][L]’1 ) = 0 in H 3 (F, Z/3Z).

The class [L ][L]’1 comes from ν ν ’1 . Since (u, ν ν ’1 ) is obviously admissible we

have a Tits construction J = J(B, „, u, ν ν ’1 ) whose invariant g3 (J ) is zero. By

Theorem (??), (??), J has zero divisors and by Theorem (??) ν ν ’1 is a norm

of B.

Let J(A, ») be a ¬rst Tits construction. Since an admissible pair for this

construction can be assumed to be of the form 1, (», »’1 ) we get

(40.11) Corollary. Let A be a central division algebra of degree 3 and let », » ∈

F — . The Albert algebras J(A, ») and J(A, » ) are isomorphic if and only if » »’1 ∈

NrdA (A— ).

536 IX. CUBIC JORDAN ALGEBRAS

We now prove the result of Parimala, Sridharan and Thakur [?] quoted in

Remark (??).

(40.12) Theorem. Let (B, „ ) be a degree 3 central simple K-algebra with a unitary

involution. Then J(B, „, u1 , ν1 ) J(B, „, u2 , ν2 ) if and only if there exists some

—

w ∈ B such that u2 = wu1 „ (w) and ν2 = ν1 NrdB (w).

Proof : Let (u1 , ν1 ), (u2 , ν2 ) be admissible pairs. Recall from (??) the equiva-

lence relation ≡ on admissible pairs. Assume that J(B, „, u1 , ν1 ) J(B, „, u2 , ν2 ).

By (??), we have some u3 such that (u1 , ν1 ) ≡ (u3 , ν2 ) and by (??) (u3 , ν2 ) ≡

u2 , NrdB (w)’1 ν2 for some w ∈ B — such that u2 = wu3 „ (w). One has NrdB (u3 ) =

ν2 ν 2 = NrdB (u3 ) since (u3 , ν2 ) and (u2 , ν2 ) are admissible pairs, thus »» = 1 for

» = NrdB (w). Let „2 = Int(u2 ) —¦ „ . By the next lemma applied to „2 , there exists

w ∈ B — such that w „2 (w ) = 1 and » = NrdB (w ). It follows from w „2 (w ) = 1

that w u2 „ (w ) = u2 , hence

u2 , NrdB (w)’1 ν2 ≡ u2 , NrdB (w ) NrdB (w)’1 ν2 = (u2 , ν2 )

and (u1 , ν1 ) ≡ (u2 , ν2 ) as claimed. The converse is (??), (??).

(40.13) Lemma (Rost). Let (B, „ ) be a degree 3 central simple K-algebra with a

unitary involution. Let w ∈ B — be such that » = NrdB (w) ∈ K satis¬es »» = 1.

Then there exists w ∈ B — such that w „ (w ) = 1 and » = NrdB (w ).

Proof : Assume that an element w as desired exists and assume that M = K[w ] ‚

B is a ¬eld. We have „ (M ) = M , so let H = M „ be the sub¬eld of ¬xed elements

under „ . The extension M/H is of degree 2 and by Hilbert™s Theorem 90 (??) we

√

may write w = u„ (u)’1 . Since M √ H — K and K = F ( a) for some a ∈ F — , we

=

may choose u of the form u = h + a with h ∈ H. Then

√ √

» = Nrd(w ) = NrdB (h + a) NrdB (h ’ a)’1 .

On the other hand » = y„ (y)’1 by Hilbert™s Theorem 90 (??), so that h ∈ H(B, „ )

is a zero of

√ √ √ ’1

•(h) = y NrdB (h ’ a) ’ y NrdB (h + a) a .

√ ’1

(the factor a is to get an F -valued function on H(B, „ )). Reversing the argu-

√ √

ment, if • is isotropic on H(B, „ ), then w = (h + a)(h ’ a)’1 is as desired. The

function • is polynomial of degree 3 and it is easily seen that • is isotropic over K.

It follows that • is isotropic over F (see Exercise ?? of this chapter).

(40.14) Remark. Suresh has extended Lemma (??) to algebras of odd degree

with unitary involution (see [?], see also Exercise 12, (b), in Chapter III).

Theorem (??) has a nice Skolem-Noether type application, which is also due

to Parimala, Sridharan and Thakur [?]:

(40.15) Corollary. Let (B1 , „1 ), (B2 , „2 ) be degree 3 central simple algebras over K

with unitary involutions. Suppose that H(B1 , „1 ) and H(B2 , „2 ) are subalgebras of

an Albert algebra J and that ± : H(B1 , „1 ) H(B2 , „2 ) is an isomorphism of Jordan

algebras. Then ± extends to an automorphism of J.

Proof : In view of Theorem (??), (??), we have isomorphisms

∼ ∼

ψ1 : J(B1 , „1 , u1 , ν1 ) ’ J,

’ ψ2 : J(B2 , „2 , u2 , ν2 ) ’ J.

’

§41. EXCEPTIONAL SIMPLE LIE ALGEBRAS 537

∼

By Proposition (??) ± extends to an isomorphism ± : (B1 , „1 ) ’ (B2 , „2 ), thus we

’

get an isomorphism of Jordan algebras

∼

J(±) : J(B1 , „1 , u1 , ν1 ) ’ J B2 , „2 , ±(u1 ), ν1 .

’

But J B2 , „2 , ±(u1 ), ν1 J(B2 , „2 , u2 , ν2 ), since both are isomorphic to J. By The-

—

orem (??), there exists w ∈ B2 such that u2 = w±(u1 )„2 (w) and ν2 = NrdB (w)ν1 .

Let

∼

φ : J B2 , „2 , ±(u1 ), ν1 ’ J(B2 , „2 , u2 , ν2 )

’

be given by (a, b) ’ (a, bw). Then φ restricts to the identity on H(B2 , „2 ) and

’1

ψ = ψ2 —¦ φ —¦ J(±) —¦ ψ1

is an automorphism of J extending ±.

40.A. Invariants of twisted compositions. Let F be a ¬eld of character-

istic not 2. To a twisted composition (V, L, Q, β) we may associate the following

cohomological invariants:

(a) a class f1 = [δ] ∈ H 1 (F, µ2 ), which determines the discriminant ∆ of L;

(b) a class g1 ∈ H 1 F, (Z/3Z)δ which determines L (with the ¬xed discrimi-

nant ∆ given by the cocycle δ);

(c) invariants f3 ∈ H 3 (F, µ2 ), f5 ∈ H 5 (F, µ2 ), and g3 ∈ H 3 (F, Z/3Z) which

are the cohomological invariants associated with the Freudenthal algebra

J(L, V ) (see Theorem (??)).

As for Freudenthal algebras, it is unknown if these invariants classify twisted com-

positions, however:

(40.16) Proposition. The invariant g3 of a twisted composition (V, L, Q, β) is

trivial if and only if (V, L, Q, β) is similar to a composition “(C, L) of type G2 , in

which case (V, L, Q, β) is classi¬ed up to similarity by f1 and g1 (which determine L)

and by f3 (which determines C).

Proof : By Theorem (??) J(V, L) has zero divisors if and only if (V, L, Q, β) is

similar to a composition of type G2 , hence the claim by Theorem (??).

§41. Exceptional Simple Lie Algebras

There exists a very explicit construction, due to Tits [?], of models for all

exceptional simple Lie algebras. This construction is based on alternative algebras

of degree 2 or 1 and Jordan algebras of degree 3 or 1. We sketch it and refer to

[?], to the book of Schafer [?] or to the notes of Jacobson [?] for more details. We

assume throughout that the characteristic of F is di¬erent from 2 and 3. Let A,

B be Hurwitz algebras over F and let J = H3 (B, ±) be the Freudenthal algebra

associated to B and ± = diag(±1 , ±2 , ±3 ). As usual we write A0 , resp. J 0 for the

trace zero elements in A, resp. J. We de¬ne a bilinear product — in A0 by

1

a — b = ab ’ 2 T (a, b)

where T (a, b) = TA (ab), a, b ∈ A, is the bilinear trace form of A. Let a , resp. ra ∈

EndF (A) be the left multiplication map a (x) = ax, resp. the right multiplication

map ra (x) = xa. For f , g ∈ EndF (A) we put [f, g] = f —¦ g ’ g —¦ f for the Lie

product in EndF (A). It can be checked that in any alternative algebra A

Da,b = [ a , b ] + [ a , rb ] + [ra , rb ]

538 IX. CUBIC JORDAN ALGEBRAS

is a derivation. Similarly we may de¬ne a product on J 0 :

x — y = xy ’ 1 T (x, y)

3

where T (x, y) = TJ (xy). We now de¬ne a bilinear and skew-symmetric product

[ , ] on the direct sum of F -vector spaces

L(A, J) = Der(A, A) • A0 — J 0 • Der(J, J)

as follows:

(1) [ , ] is the usual Lie product in Der(A, A) and Der(J, J) and satis¬es [D, D ] = 0

for D ∈ Der(A, A) and D ∈ Der(J, J),

(2) [a — x, D + D ] = D(a) — x + a — D (x) for a ∈ A0 , x ∈ J 0 , D ∈ Der(A, A) and

D ∈ Der(J, J),

1 1

(3) [a — x, b — y] = 12 T (x, y)Da,b + (a — b) — (x — y) + 2 T (a, b)[rx , ry ] for a, b ∈ B 0

and x, y ∈ J 0 .

With this product L(A, J) is a Lie algebra. As A and B vary over the possible

composition algebras the types of L(A, J) can be displayed in a table, whose last

four columns are known as Freudenthal™s “magic square”:

dim A F F —F —F H3 (F, ±) H3 (K, ±) H3 (Q, ±) H3 (C, ±)

1 0 0 A1 A2 C3 F4

2 0 A2 A2 • A 2 A5 E6

U

4 A1 A1 • A 1 • A 1 C3 A5 D6 E7

8 G2 D4 F4 E6 E7 E8

Here K stands for a quadratic ´tale algebra, Q for a quaternion algebra and C for a

e

Cayley algebra; U is a 2-dimensional abelian Lie algebra. The fact that D4 appears

in the last row is one more argument for considering D4 as exceptional.

Exercises

1. (Springer [?, p. 63]) A cubic form over a ¬eld is isotropic if and only if it is

isotropic over a quadratic extension.

2. For any alternative algebra A over a ¬eld of characteristic not 2, A + is a special

Jordan algebra.

3. Show that in all cases considered in §??, §??, and §?? the generic norm

NJ ( i xi ui ) is irreducible in F [x1 , . . . , xn ].

4. Let C be a Hurwitz algebra. Show that H2 (C, ±) is the Jordan algebra of a

quadratic form.

5. Show that a Jordan division algebra of degree 2 is the Jordan algebra J(V, q)

of a quadratic form (V, q) such that bq (x, x) = 1 for all x ∈ V .

6. Let J be a cubic Jordan structure. The following conditions are equivalent:

(a) J contains an idempotent (i.e an element e with e2 = e) such that SJ (e) =

1.

(b) J contains a nontrivial idempotent e.

(c) J contains nontrivial zero divisors.

(d) There is some nonzero a ∈ J such that NJ (a) = 0.

(e) There is some nonzero a ∈ J such that a# = 0.

7. (a) Let A be a cubic separable alternative algebra and let » ∈ F — . Check that

the norm NA induces a cubic structure J(A, ») on A • A • A.

EXERCISES 539

(b) Show that J(A, ») H3 (C, ±) for some ± if A = F — C, with C a Hurwitz

algebra over F .

8. (Rost) Let A be central simple of degree 3, » ∈ F — , and J = J(A, ») the

corresponding ¬rst Tits construction. Put:

V (J) = { [ξ] ∈ P26 | ξ ∈ J, ξ # = 0 }

F

PGL1 (A) = { x ∈ P(A) = P8 | NrdA (x) = 0 }

F

SL1 (A)» = { x ∈ A | NrdA (x) = » }.

Show that

(a) V (J) is the projective variety with coordinates [a, b, c] ∈ P(A • A • A) and

equations

a# = bc, b# = »’1 ca, c# = »ab

and V (J) is smooth.

(b) The open subvariety U of V (J) given by

NrdA (a) NrdA (b) NrdA (c) = 0

is parametrized by coordinates

[a, b] ∈ P(A • A)

with NrdA (a) = » NrdA (b) and NrdA (a) NrdA (b) = 0.

(c) SL1 (A)» — PGL1 (A) is an open subset of V (J).

(d) SL1 (A)—PGL1 (A) and SL3 (F )—PGL3 (F ) are birationally equivalent (and

rational). Hint: Use that J(A, 1) J M3 (F ), 1 .

Show that a special Jordan central division algebra over R is either isomor-

9.

phic to R or to to the Jordan algebra of a negative de¬nite quadratic form of

dimension ≥ 2 over R.

10. Let J be an Albert algebra over F . Show that:

(a) J is split if F is ¬nite or p-adic.

(b) J is reduced if F = R or if F is a ¬eld of algebraic numbers (Albert-

Jacobson [?]).

Let Ca be the nonsplit Cayley algebra over R. Show that the Albert algebras

11.

H3 (Ca , 1), H3 Ca , diag(1, ’1, 1) , and H3 Cs , diag(1, ’1, 1) are up to isomor-

phism all Albert algebras over R.

12. Let F be a ¬eld of characteristic not 2 and J = H3 (C, 1), J1 = H3 (Q, 1),

J2 = H3 (K, 1) and J3 = H3 (F, 1) for C a Cayley algebra, Q a quaternion

algebra, and K = F (i), i2 = a, a quadratic ´tale algebra. Show that

e

(a) AutF (J/J1 ) SL1 (Q).

(b) AutF (J/J2 ) SU(M, h) where M = K ⊥ ‚ C (with respect to the norm)

and

h(x, y) = NC (x, y) + a’1 iNC (ix, y).

In particular AutF (J/J2 ) SL3 (F ) if K = F — F .

(c) AutF (J/J3 ) AutF (C).

(d) AutF (J1 ) — SL1 (Q) ’ AutF (J) (“C3 — A1 ‚ F4 ”).

(e) AutF (J2 ) — SU(M, h) ’ AutF (J) (“A2 — A2 ‚ F4 ”).

(f) AutF (J3 ) — SL2 (F ) ’ AutF (J) (“G2 — A1 ‚ F4 ”).

540 IX. CUBIC JORDAN ALGEBRAS

(g) Let

±«

00 0

V = 0 x c x ∈ F, c ∈ C ‚ J.

0 c ’x

Show that AutF (J/F · E11 ) Spin9 (V, T |V ). (“B4 ‚ F4 ”).

(h) AutF (J/F · E11 • F · E22 • F · E33 ) Spin(C, n).

Observe that (??), (??), and (??) give the possible types of maximal subgroups

of maximal rank for F4 .

13. (Parimala, Sridharan, Thakur) Let J = J(B, „, uν) be a second Tits construc-

tion with B = M3 (K) and u ∈ B such that NrdB (u) = 1. Let u K be the

hermitian form of rank 3 over K determined by u and let C = C( u K , K) be

the corresponding Cayley algebra, as given in Exercise ?? of Chapter ??. Show

that the class of C is the f3 -invariant of J.

Notes

§??. The article of Paige [?] provides a nice introduction to the theory of

Jordan algebras. Jacobson™s treatise [?] gives a systematic presentation of the

theory over ¬elds of characteristic not 2. Another important source is the book

of Braun-Koecher [?] and a forthcoming source is a book by McCrimmon [?]. If

2 is not invertible the Jordan identity (??) is unsuitable and a completely new

characteristic-free approach was initiated by McCrimmon [?]. The idea is to ax-

1

iomatize the quadratic product aba instead of the Jordan product a q b = 2 (ab + ba).

McCrimmon™s theory is described for example in Jacobson™s lecture notes [?] and

[?]. Another approach to Jordan algebras based on an axiomatization of the notion

of inverse is provided in the book of Springer [?]. The treatment in degree 3 is

similar to that given by McCrimmon for exceptional Jordan algebras (see [?, §5]).

A short history of Jordan algebras can be found in Jacobson™s obituary of Albert

[?], and a survey for non-experts is given in the paper by McCrimmon [?]. For

more recent developments by the Russian School, especially on in¬nite dimensional

Jordan algebras, see McCrimmon [?].

A complete classi¬cation of ¬nite dimensional simple formally real Jordan al-

gebras over R appears already in Jordan, von Neumann and Wigner [?]35 . They

conjectured that H3 (C, 1) is exceptional and proposed it as a problem to Albert.

Albert™s solution appeared as a sequel [?] to their paper. Much later, Albert again

took up the theory of Jordan algebras; in [?] he described the structure of simple

Jordan algebras over algebraically closed ¬elds of characteristic not 2, assuming

that the algebras admit an idempotent di¬erent from the identity. (The existence

of an identity in a simple ¬nite dimensional Jordan algebra was showed by Albert

in [?].) The gap was ¬lled by Jacobson in [?]. In [?] Schafer showed that reduced

exceptional Jordan algebras of dimension 27 are all of the form H3 (C, ±). A system-

atic study of algebras H3 (C, ±) is given in Freudenthal™s long paper [?], for example

the fact that they are of degree 3. In the same paper Freudenthal showed that

the automorphism group of a reduced simple exceptional Jordan algebra over R is

of type F4 by computing the root system explicitly. In a di¬erent way, Springer

35 A a2 = 0 implies every ai = 0.

Jordan algebra is said to be formally real if i

NOTES 541

[?, Theorem 3] or [?], showed that the automorphism group is simply connected

of dimension 52, assuming that F is a ¬eld of characteristic di¬erent from 2 and

3, and deduced its type using the classi¬cation of simple algebraic groups. The

fact that the derivation algebra of an exceptional Jordan algebra is a Lie algebra

of type F4 can already be found in Chevalley-Schafer [?]. Here also it was assumed

that the base ¬eld has characteristic di¬erent from 2 and 3. Observe that this Lie

algebra is not simple in characteristic 3. Split simple groups of type E6 also occur in

connection with simple split exceptional Jordan algebras, namely as automorphism

groups of the cubic form N , see for example Chevalley-Schafer [?], Freudenthal [?]

and Jacobson [?].

The structure of algebras H3 (C, ±) over ¬elds of characteristic not 2, 3 was sys-

tematically studied by Springer in a series of papers ([?], [?], [?], and [?]). Some of

the main results are the fact that the bilinear trace form and C determine H3 (C, ±)

(Theorem (??), see [?, Theorem 1, p. 421]) and the fact that the cubic norm deter-

mines C (see [?, Theorem 1]). Thus the cubic norm and the trace form determine

the algebra. The fact that the trace form alone determines the algebra was only

recently noticed by Serre and Rost (see [?, § 9.2]). The fact that the isomorphism

class of C is determined by the isomorphism class of H3 (C, ±) is a result due to

Albert-Jacobson [?]. For this reason C is usually called the coordinate algebra of

H3 (C, ±). A recent survey of the theory of Albert algebras has been given by

Petersson and Racine [?].

It is unknown if a division Albert algebra J always contains a cyclic cubic

¬eld extension (as does an associative central simple algebra of degree 3). However

this is true (Petersson-Racine [?, Theorem 4]) if char F = 3 and F contains a

primitive cube root of unity: it su¬ces to show that J contains a Kummer extension

F [X]/(X 3 ’ »), hence that SJ restricted to J 0 = { x ∈ J | TJ (x) = 0 } is isotropic.

In view of Springer™s theorem, we may replace J by J — L where L is a cubic ´tale e

0

subalgebra of J. But then J is reduced and then SJ |J is isotropic.

§??. There are a number of characterizations of cubic Jordan algebras. One

is due to Springer [?], assuming that char F = 2, 3: Let J be a ¬nite dimensional

F -algebra with 1, equipped with a quadratic form Q such that

(a) Q(x)2 = Q(x2 ) if bQ (x, 1) = 0,

(b) bQ (xy, z) = bQ (x, yz),

3

(c) Q(1) = 2 .

1

Then J is a cubic Jordan algebra and Q(x) = 2 TJ (x2 ).

The characterization we use in §?? was ¬rst suggested by Freudenthal [?] and

was established by Springer [?] for ¬elds of characteristic not 2 and 3. We follow the

description of McCrimmon [?], which is systematically used by Petersson-Racine

in their study of cubic Jordan algebras (see for example [?] and [?]). The Springer

decomposition is given in the G¨ttinger notes of Springer [?]. Applications were

o

given by Walde [?] to the construction of exceptional Lie algebras. The construction

was formalized and applied to cubic forms by Petersson and Racine (see for example

[?]).

§??. Tits constructions for ¬elds of characteristic not 2 ¬rst appeared in print

in Jacobson™s book [?], as did the fact, also due to Tits, that any Albert algebra is a

¬rst or second Tits construction. These results were announced by Tits in a talk at

542 IX. CUBIC JORDAN ALGEBRAS

the Oberwolfach meeting “Jordan-Algebren und nicht-assoziativen Algebren, 17“

26.8.1967”. With the kind permission of J. Tits and the Research Institute in

Oberwolfach, we reproduce Tits™ R´sum´:

e e

Exceptional simple Jordan Algebras

(I) Denote by k a ¬eld of characteristic not 2, by A a central simple algebra of

degree 3 over k, by n : A ’ A, tr : A ’ A the reduced norm and reduced trace,

and by — : A—A ’ A the symmetric bilinear product de¬ned by (x—x)x = n(x).

For x ∈ A, set x = 1 (tr(x) ’ x). Let c ∈ k — . In the sum A0 + A1 + A2 of three

2

copies of A, introduce the following product:

x0 y1 z2

1

(xx + x x)0 (x y)1 (zx )2

x0 2

(xy )1 c(y — y )2 (y z)0

y1

1

(z x)2 (yz )0 (z — z )1

z2 c

(II) Denote by a quadratic extension of k, by B a central simple algebra of

degree 3 over , and by σ : B ’ B an involution of the second kind kind such

that k = { x ∈ | xσ = x }. Set B Sym = { x ∈ B | xσ = x }. Let b ∈ B Sym and

c ∈ l— be such that n(b) = cσ c. In the sum B Sym + B— of B Sym and a copy B—

of B, de¬ne a product by

x y

1

(xx + x x) (x y)—

x 2

σ

+ y by σ )+ cσ (y σ — y )b’1

σ

(xy )— (yby

y —

Theorem 1. The 27-dimensional algebras described under (I) and (II) are ex-

ceptional simple Jordan algebras over k. Every such algebra is thus obtained.

Theorem 2. The algebra (I) is split if c ∈ n(A) and division otherwise. The

algebra (II) is reduced if c ∈ n(B) and division otherwise.

Theorem 3. There exists an algebra of type (II) which does not split over any

cyclic extension of degree 2 or 3 of k. (Notice that such an algebra is necessarily

division and is not of type (I)).

(For more details, cf. N. Jacobson. Jordan algebras, a forthcoming book).

J. Tits

Observe that the —-product used by Tits is our —-product divided by 2. The

extension of Tits constructions to cubic structures was carried out by McCrimmon

[?]. Tits constructions were systematically used by Petersson and Racine, see for

example [?] and [?]. Petersson and Racine showed in particular that (with a few

exceptions) simple cubic Jordan structures can be constructed by iteration of the

Tits process ([?], Theorem 3.1). The result can be viewed as a cubic analog to the

theorem of Hurwitz, proved by iterating the Cayley-Dickson process.

Tits constructions can be used to give simple examples of exceptional division

Jordan algebras of dimension 27. The ¬rst examples of such division algebras were

constructed by Albert [?]. They were signi¬cantly more complicated than those

through Tits constructions. Assertions (??) and (??) of Theorem (??) and (??) are

NOTES 543

due to Tits. The nice cohomological proof given here is due to Ferrar and Petersson

[?] (for ¬rst Tits constructions).

§??. The existence of the invariants f3 and f5 was ¬rst noticed by Serre (see

for example [?]). The direct computation of the trace form given here, as well as

Propositions (??) and (??) are due to Rost. Serre suggested the existence of the

invariant g3 . Its de¬nition is due to Rost [?]. An elementary approach to that

invariant can be found in Petersson-Racine [?] and a description in characteristic 3

can be found in Petersson-Racine [?].

544 IX. CUBIC JORDAN ALGEBRAS

CHAPTER X

Trialitarian Central Simple Algebras

We assume in this chapter that F is a ¬eld of characteristic not 2. Triality

for PGO+ , i.e., the action of S3 on PGO+ and its consequences, is the subject of

8 8

this chapter. In the ¬rst section we describe the induced action on H 1 (F, PGO+ ).8

This cohomology set classi¬es ordered triples (A, B, C) of central simple algebras of

∼

degree 8 with involutions of orthogonal type such that C(A, σA ), σ ’ (B, σB ) —

’

(C, σC ). Triality implies that this property is symmetric in A, B and C, and the

induced action of S3 on H 1 (F, PGO+ ) permutes A, B, and C. As an application we

8

give a criterion for an orthogonal involution on an algebra of degree 8 to decompose

as a tensor product of three involutions.

We may view a triple (A, B, C) as above as an algebra over the split ´tale e

algebra F — F — F with orthogonal involution (σA , σB , σC ) and some additional

∼

structure (the fact that C(A, σA ), σ ’ (B, σB ) — (C, σC )). Forms of such “alge-

’

bras”, called trialitarian algebras, are classi¬ed by H 1 (F, PGO+ S3 ). Trialitarian

8

algebras are central simple algebras with orthogonal involution of degree 8 over cu-

bic ´tale F -algebras with a condition relating the central simple algebra and its

e

Cli¬ord algebra. Connected components of automorphism groups of such trialitar-

ian algebras give the outer forms of simple adjoint groups of type D4 .

Trialitarian algebras also occur in the construction of Lie algebras of type D 4 .

Some indications in this direction are in the last section.

§42. Algebras of Degree 8

42.A. Trialitarian triples. The pointed set H 1 (F, PGO8 ) classi¬es central

simple algebras of degree 8 over F with an involution of orthogonal type and

the image of the pointed set H 1 (F, PGO+ ) in H 1 (F, PGO8 ) classi¬es central

8

simple algebras of degree 8 over F with an involution of orthogonal type hav-

ing trivial discriminant (see Remark (??)). More precisely, each cocycle x in

H 1 (F, PGO+ ) determines a central simple F -algebra A(x) of degree 8 with an

8

orthogonal involution σA(x) having trivial discriminant, together with a designa-

tion of the two components C + A(x), σA(x) and C ’ A(x), σA(x) of the Cli¬ord

algebra C A(x), σA(x) . Thus, putting B(x), σB(x) = C + A(x), σA(x) , σ and

C(x), σC(x) = C ’ A(x), σA(x) , σ , x determines an ordered triple

A(x), σA(x) , B(x), σB(x) , C(x), σC(x)

of central simple F -algebras of degree 8 with orthogonal involution. The two com-

ponents of the Cli¬ord algebra C A(x), σA(x) are determined by a nontrivial cen-

tral idempotent e, say B(x) = C A(x), σA(x) e and C(x) = C A(x), σA(x) (1 ’

e). Thus two triples [(A, σA ), (B, σB ), (C, σC )] and [(A , σA ), (B , σB ), (C , σC )],

where B = C(A, σA )e and C = C(A, σA )(1 ’ e), resp. B = C(A , σA) )e and

545

546 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

C = C(A , σA )(1 ’ e ), determine the same class in H 1 (F, PGO+ ) if there exists

8

an isomorphism φ : (A, σA ) ’ (A , σA ) such that C(φ)(e) = e . Now let (A, B, C)

be an ordered triple of central simple algebras of degree 8 with orthogonal involution

such that there exists an isomorphism

∼

±A : C(A, σA ), σ ’ (B, σB ) — (C, σC ).

’

The element e = ±’1 (0, 1) is a central idempotent of (A, σA ), hence determines

A

a designation of the two components of C(A, σA ). Moreover this designation is

independent of the particular choice of ±A , since it depends only on the ordering of

the triple. We call two triples (A, B, C) and (A , B , C ) isomorphic if there exist

isomorphisms of algebras with involution

∼ ∼ ∼

(φ1 : A ’ A , φ2 : B ’ B , φ3 : C ’ C )

’ ’ ’

and isomorphisms ±A , resp. ±A as above, such that

±A —¦ C(φ1 ) = (φ2 , φ3 ) —¦ ±A .

Thus:

(42.1) Lemma. The set H 1 (F, PGO+ ) classi¬es isomorphism classes of ordered

8

triples (A, B, C) of central simple F algebras of degree 8 with involutions of orthog-

onal type and trivial discriminant.

Observe that the ordered triples (A, B, C) and (A, C, B) determine in general

di¬erent objects in H 1 (F, PGO+ ) since they correspond to di¬erent designations

8

of the components of C(A, σA ). In fact the action of S2 on H 1 (F, PGO+ ) induced

8

by the exact sequence of group schemes

d

1 ’ PGO+ (A, σ) ’ PGO(A, σ) ’ S2 ’ 1

’

permutes the classes of (A, B, C) and (A, C, B).

(42.2) Example. Let A1 = EndF (C) and σ1 = σn where C is a split Cayley

algebra with norm n. In view of proposition (??) we have a canonical isomorphism

±C : C(A1 , σ1 ), σ ’ (A2 , σ2 ) — (A3 , σ3 )

where (A2 , σ2 ) and (A3 , σ3 ) are copies of the split algebra (A1 , σ1 ). Thus the ordered

triple (A1 , A2 , A3 ) determines a class in H 1 (F, PGO+ ). Since n is hyperbolic, it

8

corresponds to the trivial class.

The group S3 acts as outer automorphisms on the group scheme PGO+ (see

8

+

1

Proposition (??)). It follows that S3 acts on H (F, PGO8 ).

(42.3) Proposition. The action of S3 on H 1 (F, PGO+ ) induced by the action

8

+

of S3 on PGO8 is by permutations on the triples (A, B, C). More precisely, the

choice of an isomorphism

∼

±A : C(A, σA ), σ ’ (B, σB ) — (C, σC )

’

determines isomorphisms

∼

±B : C(B, σB ), σ ’ (C, σC ) — (A, σA ),

’

∼

±C : C(C, σC ), σ ’ (A, σA ) — (B, σB ).

’

Moreover any one of the three ±A , ±B or ±C determines the two others.

§42. ALGEBRAS OF DEGREE 8 547

Proof : We have PGO+ (Fsep ) = PGO+ (Cs , ns )(Fsep ) and we can use the de-

8

scription of the action of S3 on PGO+ (Cs , ns )(Fsep ) given in Proposition (??).

Let θ and θ± be the automorphisms of PGO+ (Cs , ns ) as de¬ned in (??). Let

x = (γg )g∈Gal(Fsep /F ) with γg = [tg ], tg ∈ O+ (Fsep ), be a cocycle in H 1 (F, PGO+ )

8 8

+

+’ +’

which de¬nes (A, σA ). By de¬nition of (θ , θ ) the map (θ , θ ) : PGO ’

PGO+ — PGO+ factors through Autalg C0 (n), σ — Autalg C0 (n), σ . Hence

+ ’

the cocycle θ + x = θ+ ([tg ]) de¬nes the triple (B, σB , ±B ) and θ’ x = θ’ ([tg ]) de-

¬nes (C, σC , ±C ). The last assertion follows by triality.

(42.4) Example. In the situation of Example (??), where A1 = A2 = A3 =

EndF (C) and ±A1 = ±C we obtain ±A2 = ±A3 = ±C since the trivial cocycle

represents the triple (A1 , A2 , A3 ).

We call an ordered triple (A, B, C) of central simple algebras of degree 8 such

∼

that there exists an isomorphism ±A : C(A, σA ), σ ’ (B, σB ) — (C, σC ) a triali-

’

tarian triple. For any φ ∈ S3 , we write the map ± induced from ±A by the action

+

of S3 as ±φ . For example we have ±θ = ±B .

A

A

(42.5) Proposition. Let (A, B, C) be a trialitarian triple. Triality induces iso-

morphisms

Spin(A, σA ) Spin(B, σB ) Spin(C, σC ),

PGO+ (A, σA ) PGO+ (B, σB ) PGO+ (C, σC ).

Proof : Let γ = γg = [tg ], tg ∈ GO+ (Fsep ), be a 1-cocycle de¬ning (A, σA , ±A ) so

8

that γ + = θ+ γ de¬nes (B, σB , ±B ). Since Int PGO+ (Fsep ) = PGO+ (Fsep ) we

8 8

+

may use γ to twist the Galois action on PGO8 . The isomorphism

θ+ : PGO+ ’ (PGO+ )γ

8 8

then is a Galois equivariant map, which descends to an isomorphism

∼

PGO+ (A, σA ) ’ PGO+ (B, σB ).

’

The existence of an isomorphism between corresponding simply connected groups

then follows from Theorem (??).

(42.6) Remark. If (A, σ) is central simple of degree 2n with an orthogonal invo-

lution, the space

L(A, σ) = { x ∈ A | σ(x) = ’x }

of skew-symmetric elements is a Lie algebra of type Dn under the product [x, y] =

xy ’ yx (since it is true over a separable closure of F , see [?, Theorem 9, p. 302]).

In fact L(A, σ) is the Lie algebra of the groups Spin(A, σ) or PGO+ (A, σ) (see

??), so that Proposition (??) implies that

L(A, σA ) L(B, σB ) L(C, σC )

∼

if A is of degree 8 and C(A, σA ) ’ (B, σB ) — (C, σC ). An explicit example where

’

(A, σA ) (B, σB ), but L(A, σA ) L(B, σB ) is in Jacobson [?, Exercise 3, p. 316].

(42.7) Proposition. Let (A, B, C) be a trialitarian triple. We have

(1) [A][B][C] = 1 in Br(F ).

(2) A EndF (V ) if and only if B C.

548 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

(3) A B C if and only if (A, σA ) EndF (C), σn for some Cayley algebra C

with norm n.

Proof : (??) is a special case of Theorem (??), see also Example (??), and (??) is

an immediate consequence of (??).

The “if” direction of (??) is a special case of Proposition (??). For the converse,

it follows from [A] = [B] = [C] = 1 in Br(F ) that (A, σA ) = EndF (V ), σq

and that (V, q) has trivial discriminant and trivial Cli¬ord invariant. In view of

Proposition (??) (V, q) is similar to the norm n of a Cayley algebra C over F . This

implies (A, σA ) EndF (C), σn .

(42.8) Remark. As observed by A. Wadsworth [?], there exist examples of tri-

alitarian triples (A, B, C) such that all algebras A, B, C are division algebras:

Since there exist trialitarian triples EndF (V ), B, B with B a division algebra (see

Dherte [?], Tao [?], or Yanchevski˜ [?]), taking B to be generic with an involution

±

of orthogonal type and trivial discriminant (see Saltman [?]) will give such triples.

42.B. Decomposable involutions. We consider central simple F -algebras

of degree 8 with involutions of orthogonal type which decompose as a tensor prod-

uct of three involutions. In view of Proposition (??) such involutions have trivial

discriminant.

(42.9) Proposition. Let A be a central simple F -algebra of degree 8 with an in-

volution σ of orthogonal type. Then (A, σ) (A1 , σ1 ) — (A2 , σ2 ) — (A3 , σ3 ) with Ai ,

i = 1, 2, 3, quaternion algebras and σi an involution of the ¬rst kind on Ai , if and

only if (A, σ) is isomorphic to C(q0 ), „ where C(q0 ) is the Cli¬ord algebra of a

quadratic space (V0 , q0 ) of rank 6 and „ is the involution which is ’Id on V0 .

Proof : We ¬rst check that the Cli¬ord algebra C(q0 ) admits such a decomposition.

Let q0 = q4 ⊥ q2 be an orthogonal decomposition of q0 with q4 of rank 4 and q2 of

rank 2. Accordingly, we have a decomposition C(q0 ) = C(q4 ) — C(q2 ) where — is

the Z/2Z-graded tensor product (see for example Scharlau [?, p. 328]). Let z be a

generator of the center of C0 (q4 ) such that z 2 = δ4 ∈ F — . The map

φ(x — 1 + 1 — y) = x — 1 + z — y

induces an isomorphism

∼

C(q0 ) = C(q4 ) — C(q2 ) ’ C(δ4 q4 ) — C(q2 )

’

by the universal property of the Cli¬ord algebra. The canonical involution of C(q 0 )

is transported by φ to the tensor product of the two canonical involutions, since z

is invariant by the canonical involution of C(q4 ). Similarly, we may decompose q4

as q4 = q ⊥ q and write

∼

C(q4 ) = C(q ) — C(q ) ’ C(q ) — C(δ q )

’

where δ is the discriminant of q . In this case the canonical involution of C(q )

2

maps a generator z of the center of C0 (q ) such that z = δ ∈ F — to ’z . We

then have to replace the canonical involution of C(q ) (which is of orthogonal type)

by the “second” involution of C(q ), i.e., the involution „ such that „ (x) = ’x

on V . This involution is of symplectic type. Conversely, let

(A, σ) (A1 , σ1 ) — (A2 , σ2 ) — (A3 , σ3 ).

Renumbering the algebras if necessary, we may assume that σ1 is of orthogonal type

and that there exists a quadratic space (V1 , q1 ) such that (A1 , σ1 ) C(q1 ), „1 with

§42. ALGEBRAS OF DEGREE 8 549

„1 the canonical involution of C(q1 ). We may next assume that σ2 and σ3 are of

symplectic type: if σ2 and σ3 are both of orthogonal type, σ2 — σ3 is of orthogonal

type and has trivial discriminant by Proposition (??). Corollary (??) implies that

(A2 , σ2 ) — (A3 , σ3 ) (B, σB ) — (C, σC )

where σB , σC are the canonical involutions of the quaternion algebras B, C, and

we replace (A2 , σ2 ) by (B, σB ), (A3 , σ3 ) by (C, σC ). Then there exist quadratic

forms q2 , q3 such that (A2 , σ2 ) C(q2 ), „2 and (A3 , σ3 ) C(q3 ), „3 , with „2 ,

„3 “second involutions”, as described above. Let δi be the discriminant of qi and

let q0 = q3 ⊥ δ3 q2 ⊥ δ3 δ2 q1 , then (A, σ) C(q0 ), „ .

Algebras C(q0 ), „ occur as factors in special trialitarian triples:

(42.10) Proposition. A triple EndF (V ), B, B is trialitarian if and only if

(B, σB ) C(V0 , q0 ), „ ,

where „ is the involution of C(q0 ) which is ’Id on V0 , for some quadratic space

(V0 , q0 ) of dimension 6.

Proof : Let (A, σ) = EndF (V ), σq be split of degree 8, so that C(A, σ) = C0 (q),

and assume that q has trivial discriminant. Replacing q by »q for some » ∈ F — ,

if necessary, we may assume that q represents 1. Putting q = 1 ⊥ q1 , we de¬ne

∼

an isomorphism ρ : C(’q1 ) ’ C0 (q) by ρ(x) = xv1 where v1 is a generator of 1 .

’

Since the center Z of C0 (q) splits and since C(’q1 ) Z — C0 (’q1 ), we may

∼

’1

view ρ as an isomorphism C0 (q) ’ C0 (’q1 ) — C0 (’q1 ). The center of C0 (q) is

’

¬xed under the canonical involution of C0 (q) since 8 ≡ 0 mod 4. Thus, with the

canonical involution on all three algebras, ρ’1 is an isomorphism of algebras with

involution and the triple

EndF (V ), C0 (’q1 ), C0 (’q1 )

is a trialitarian triple. Let ’q1 = a ⊥ q2 with q2 of rank 6 and let q0 = ’aq2 , then

C0 (’q1 ) C(q0 ) as algebras with involution where the involution on C(q0 ) is the

“second involution”. Thus the triple EndF (V ), C(q0 ), C(q0 ) is trialitarian.

We now characterize fully decomposable involutions on algebras of degree 8:

(42.11) Theorem. Let A be a central simple F -algebra of degree 8 and σ an in-

volution of orthogonal type on A. The following conditions are equivalent:

(1) (A, σ) (A1 , σ1 ) — (A2 , σ2 ) — (A3 , σ3 ) for some quaternion algebras with invo-

lution (Ai , σi ), i = 1, 2, 3.

(2) The involution σ has trivial discriminant and there exists a trialitarian triple

EndF (V ), A, A .

(3) The involution σ has trivial discriminant and one of the factors of C(A, σ)

splits.

Proof : The algebra (A, σ) decomposes if and only if (A, σ) C(q0 ) by Lemma

(??). Thus the equivalence of (??) and (??) follows from (??).

The equivalence of (??) and (??) follows from the fact that S3 operates through

permutations on trialitarian triples.

(42.12) Remark. (Parimala) It follows from Theorem (??) that the condition

[A][B][C] = 1 ∈ Br(F )

550 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

for a trialitarian triple is necessary but not su¬cient. In fact, there exist a ¬eld F

and a central division algebra B of degree 8 with an orthogonal involution over F

which is not a tensor product of three quaternion algebras (see Amitsur-Rowen-

Tignol [?]). That such an algebra always admits an orthogonal involution with

trivial discriminant follows from Parimala-Sridharan-Suresh [?]. Thus, by Theo-

rem (??) there are no orthogonal involutions on B such that M8 (F ), B, B is a

trialitarian triple.

(42.13) Remark. If (A, σ) is central simple with an orthogonal involution which

is hyperbolic, then disc(σ) is trivial and one of the factors of the Cli¬ord algebra

C(A, σ) splits (see (??)). These conditions are also su¬cient for A to have an

orthogonal hyperbolic involution if A has degree 4 (see Proposition (??)) but they

are not su¬cient if A has degree 8 by Theorem (??).

§43. Trialitarian Algebras

43.A. A de¬nition and some properties. Let L be a cubic ´tale F -algebra.

e

We call an L-algebra D such that D — F A — B — C with A , B , C central

simple over F for every ¬eld extension F /F which splits L a central simple L-

algebra. For example any trialitarian triple (A, B, C) is a central simple L-algebra

with an involution of orthogonal type over the split cubic algebra L = F — F — F .

Conversely, let L be a cubic ´tale F -algebra and let E be a central simple algebra of

e

degree 8 with an involution of orthogonal type over L. We want to give conditions

on E/L such that E de¬nes a trialitarian triple over any extension which splits L.

Such a structure will be called a trialitarian algebra. In view of the decomposition

L — L L — L — ∆ where ∆ is the discriminant algebra of L (see (??)), we obtain

a decomposition

(E, σ) — L (E, σ) — (E2 , σ2 )

and (E2 , σ2 ) is an (L — ∆)-central simple algebra with involution of degree 8 over

L—∆, in particular is an L-algebra through the canonical map L ’ L—∆, ’ —1.

As a ¬rst condition we require the existence of an isomorphism of L-algebras with

involution

∼

±E : C(E, σ), σ ’ (E2 , σ2 ).

’

Fixing a generator ρ ∈ Gal(L — ∆/∆), this is equivalent by Corollary (??) to giving

an isomorphism of L-algebras with involution

∼

±E : C(E, σ), σ ’ ρ (E — ∆, σ — 1)

’

where ρ (E — ∆, σ — 1) denotes (E — ∆, σ — 1) with the action of L — ∆ twisted

through ρ. An isomorphism

∼

¦ : T = (E, L, σ, ±E ) ’ T = (E , L , σ , ±E )

’

∼

of such “data” is a pair (φ, ψ) where ψ : L ’ L is an isomorphism of F -algebras,

’

∼ ∼

∆(ψ) : ∆(L) ’ ∆(L ) is the induced map of discriminant algebras and φ : E ’ E

’ ’

is ψ-semilinear, such that

φ —¦ σ = σ —¦ φ and φ — ∆(ψ) —¦ ±E = ±E —¦ C(φ).

§43. TRIALITARIAN ALGEBRAS 551

(43.1) Remark. The de¬nition of ±E depends on the choice of a generator ρ of

the group Gal(L — ∆/∆) and such a choice is in fact part of the structure of T .

Since 1 — ι is an isomorphism

2

∼

ρ

(E — ∆) ’ ρ (E — ∆),

’

there is a canonical way to change generators.

If L = F — F — F is split, then

(E, σ) = (A, σA ) — (B, σB ) — (C, σC )

with (A, σA ), (B, σB ), (C, σC ) algebras over F of degree 8 with orthogonal involu-

tions and

(B — C, C — A, A — B) or

∼

ρ

E — ∆(L) ’

’

(C — B, A — C, B — A),

respectively, according to the choice of ρ. Thus an isomorphism ±E is a triple of

isomorphisms

(B — C, C — A, A — B) or

∼

(±A , ±B , ±C ) : C(A, σA ), C(B, σB ), C(C, σC ) ’

’

(C — B, A — C, B — A),

respectively. Given one of the isomorphisms ±A , ±B , or ±C , there is by Propo-

sition (??) a “canonical” way to obtain the two others, hence to extend it to an

isomorphism ±E . We write such an induced isomorphism as ±(A,B,C) and we say

that a datum

T = (A — B — C , F — F — F, σ , ± )

isomorphic to

T = A — B — C, F — F — F, (σA , σB , σC ), ±(A,B,C)

is a trialitarian F -algebra over F — F — F or that ± = ±(A,B,C) is a trialitarian

isomorphism. If L is not necessarily split, T = (E, L, σ, ±) is a trialitarian algebra

over L if over any ¬eld extension F /F which splits L, i.e., L — F F — F — F,

T — F is isomorphic to a trialitarian algebra over F — F — F .

(43.2) Example. Let (C, n) be a Cayley algebra over F and let A = EndF (C). By

Proposition (??), we have an isomorphism

∼

±C : C0 (C, n) = C(A, σn ) ’ (A, σn ) — (A, σn ),

’

which, by Proposition (??), extends to de¬ne a trialitarian structure

T = (A — A — A, F — F —, σn — σn — σn , ±C )

on the product A — A — A. More precisely, if ±C (x) = (x+ , x’ ) ∈ A — A, we may

take

(43.3) ±C (x, y, z) = (y+ , z’ ), (z+ , x’ ), (x+ , y’ )

as a trialitarian isomorphism, in view of Example (??). It corresponds to the action

ρ on (F — F )3 given by (xi , yi ) ’ (xi+1 , yi+2 ), i = 1, 2, 3 (mod 3). We say that

such a trialitarian algebra T is of type G2 and write it End(C). If C = Cs is split,

T = Ts is the split trialitarian algebra. Triality induces an action of S3 on Ts .

552 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

Assume that L/F is cyclic with generator ρ of the Galois group. The isomor-

phism

∼

x — y ’ xy, xρ(y), xρ2 (y)

L — L ’ L — L — L,

’

∼ ∼

induces an isomorphism ∆ ’ L — L and any ±E : C(E, σ) ’ ρ (E — ∆) can be

’ ’

viewed as an isomorphism

2

∼

±E : C(E, σ) ’ ρE — ρ E.

’

2

Thus (E, ρE, ρ E) is a trialitarian triple over L — L — L and by Proposition (??) ±E

determines an isomorphism

2

∼

± ρE : C(ρE, σ) ’ ρ E — E.

’

The isomorphism ± ρE is (tautologically) also an isomorphism

2

∼

C(E, σ) ’ ρE — ρ E.

’

ρ’1

We denote it by ± ρE .

ρ’1

(43.4) Proposition. The isomorphism ±E is trialitarian if and only if ± ρE =

±E .

Proof : It su¬ces to check the claim for a trialitarian triple (A, B, C), where it is

straightforward.

For trialitarian algebras over arbitrary cubic ´tale algebras L we have:

e

∼

(43.5) Corollary. An isomorphism ±E : C(E, σ) ’ ρ (E — ∆) extends to an iso-

’

morphism

2

∼

±E—∆ : C(E — ∆, σ — 1) ’ ρ (E — ∆) — ρ (E — ∆)

’

ρ’1

and ±E is trialitarian if and only if ±ρ(E—∆) = ±E—∆ .

The norm map

NL/F : Br(L) ’ Br(F ),

de¬ned for ¬nite separable ¬eld extensions L/F can be extended to ´tale F -algebras

e

L: if L = L1 — · · · — Lr where Li /F , i = 1, . . . , r, are separable ¬eld extensions

and if A = A1 — . . . Ar is L-central simple (i.e., Ai is central simple over Li ), then,

for [A] ∈ Br(L) = Br(L1 ) — · · · — Br(Lr ), we de¬ne

NL/F ([A]) = [A1 ] · . . . · [Ar ] ∈ Br(F ).

(43.6) Proposition. For any trialitarian algebra T = (E, L, σ, ±E ) the central

simple L-algebra E satis¬es NL/F ([E]) = 1 ∈ Br(F ).

Proof : The algebra C(E, σ) is L—∆-central simple and L—∆ is ´tale. We compute

e

the class of NL—∆/F C(E, σ) in the Brauer group Br(F ) in two di¬erent ways: on

one hand, by using that [NL—∆/L C(E, σ) ] = [E] in Br(L) (see Theorem (??) or

Example (??)), we see that

[NL—∆/F C(E, σ) ] = [NL/F —¦ NL—∆/L C(E, σ) ]

= [NL/F (E)]

§43. TRIALITARIAN ALGEBRAS 553

and on the other hand we have

[NL—∆/F C(E, σ) ] = [N∆/F NL—∆/∆ C(E, σ) ]

= [N∆/F NL—∆/∆ ρ (E — ∆) ]

= [N∆/F NL/F (E) — ∆ ]

= [NL/F (E)]2 ,

so that, as claimed [NL/F (E)] = 1.

(43.7) Example. A trialitarian algebra can be associated to any twisted compo-

sition “ = (V, L, Q, β): Let ρ be a ¬xed generator of the cyclic algebra L — ∆/∆,

∆ the discriminant of L. By Proposition (??) there exists exactly one cyclic com-

position (with respect to ρ) on (V, L, Q, β) — ∆. By Proposition (??) we then have

an isomorphism

∼

±V : C0 (V, Q) = C EndL (V ), σQ ’ ρ EndL (V ) — ∆ .

’

We claim that the datum EndL (V ), L, σQ , ±V is a trialitarian algebra. By

descent it su¬ces to consider the case where “ = C is of type G2 . Then the claim

follows from Example (??).

We set End(“) for the trialitarian algebra associated to the twisted composition

“.

43.B. Quaternionic trialitarian algebras. The proof of Proposition (??)

shows that the sole existence of a map ±E implies that NL/F ([E]) = 1. In fact,

the condition NL/F ([E]) = 1 is necessary for E to admit a trialitarian structure,

but not su¬cient, even if L is split, see Remark (??). We now give examples where

the condition NL/F ([E]) = 1 is su¬cient for the existence of a trialitarian structure

on E.

(43.8) Theorem. Let Q be a quaternion algebra over a cubic ´tale algebra L.

e

Then M4 (Q) admits a trialitarian structure T (Q) if and only if NL/F ([Q]) = 1 in

Br(F ).

Before proving Theorem (??) we observe that over number ¬elds any central

simple algebra which admits an involution of the ¬rst kind is of the form Mn (Q)

for some quaternion algebra Q (Albert, [?, Theorem 20, p. 161]). Thus, for such

¬elds, the condition NL/F ([E]) = 1 is necessary and su¬cient for E to admit a

trialitarian structure (see Allison [?] and the notes at the end of the chapter).

The ¬rst step in the proof of Theorem (??) is the following reduction:

(43.9) Proposition. Let L/F be a cubic ´tale algebra and let Q be a quaternion

e

algebra over L. The following conditions are equivalent:

(1) NL/F ([Q]) = 1.

(2) Q (a, b)L with b ∈ F — and NL (a) = 1.

Proof : (??) ’ (??) follows from the projection (or transfer) formula (see for ex-

ample Brown [?, V, (3.8)]). For the proof of (??) ’ (??) it su¬ces to show

(a, b)L with b ∈ F — : The condition NL/F ([Q]) = 1 then implies

that Q

√

NL (a) = NF (√b) (z) for some z ∈ F ( b), again by the projection formula. Replac-

ing a by a3 NF (√b) (z)’1 gives a as wanted. We ¬rst consider the case L = F —K, K

quadratic ´tale. Let Q1 — Q2 be the corresponding decomposition of Q. The condi-

e

tion NL/F ([Q]) = 1 is equivalent with NK/F ([Q2 ]) = [Q1 ] or NK/F (Q2 ) M2 (Q1 ).

554 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

In this case the claim follows from Corollary (??). Let now Q = (±, β)L , for L a

¬eld. We have to check that the L-quadratic form q = ±, β, ’±β represents a

nonzero element of F . Let L = F (θ) and q(x) = q1 (x) + q2 (x)θ + q3 (x)θ2 with qi

quadratic forms over F . In view of the case L = F — K, q2 and q3 have a nontrivial

common zero over L, hence the claim by Springer™s theorem for pairs of quadratic

forms (see P¬ster [?, Corollary 1.1, Chap. 9]).

(43.10) Remark. Proposition (??) in the split case L = F — F — F reduces to

the classical result of Albert that the condition [Q1 ][Q2 ][Q3 ] = 1 for quaternions

algebras Qi over F is equivalent to the existence of a, b, c such that [Q1 ] = (a, b)F ,

[Q2 ] = (a, c)F , [Q3 ] = (a, bc)F . In particular, the algebras Qi have a common

quadratic subalgebra (see Corollary (??)). Thus (??) can be viewed as a “twisted”

version of Albert™s result.

Theorem (??) now is a consequence of the following:

(43.11) Proposition. Let K/F be quadratic ´tale, let L/F be cubic ´tale and let

e e

—

a ∈ L be such that NL (a) = 1. Let Q be the quaternion algebra (K — L/L, a)L

and let E(a) = M4 (Q). There exists a trialitarian structure T = E(a), L, σ, ±

on E(a).

The main step in the construction of T is a result of Allen and Ferrar [?]. To

describe it we need some notations. Let (C, n) be the split Cayley algebra with

norm n. The vector space C has a basis (u1 , . . . , u8 ) (use Exercise 5 of Chapter ??)

such that

(a) the multiplication table of C is

u1 u2 u3 u4 u5 u6 u7 u8

u1 0 u7 ’u6 u1 ’u8 0 0 0

u2 ’u7 0 u5 u2 0 ’u8 0 0

u3 u6 ’u5 0 u3 0 0 ’u8 0

u4 0 0 0 u4 u5 u6 u7 0

u5 ’u4 0 0 0 0 u3 ’u2 u5

u6 0 ’u4 0 0 ’u3 0 u1 u6

u7 0 0 ’u4 0 u2 ’u1 0 u7

u8 u1 u2 u3 0 0 0 0 u8

(b) 1 = u4 + u8 and the conjugation map π is given by π(ui ) = ’ui for i = 4, 8

and π(u4 ) = u8 .

(c) bn (ui , uj ) = δi+4,j , i + 4 being taken mod 8, in particular {u1 , . . . , u4 } and

{u5 , . . . , u8 } span complementary totally isotropic subspaces of C.

(43.12) Lemma. Let a1 , a2 , a3 ∈ F — be such that a1 a2 a3 = 1, let

Ai = diag(ai , ai , ai , a’1 ), Bi = diag(1, 1, 1, a’1 )

i+2 i+1

in M4 (F ) and let ti = Bi Ai ∈ M8 (F ), i = 1, 2, 3. Also, write ti for the F -vector

0

0

space automorphism of C induced by ti with respect to the basis (u1 , . . . , u8 ). Then

ti is a similitude of (C, n) with multiplier ai such that

(1) a1 t1 (x y) = t2 (x) t3 (y) where is the multiplication in the para-Cayley algebra

C.

(2) ti ∈ Sym End(C), σn , in particular t2 = ai · 1, i = 1, 2, 3.

i

Proof : A lengthy computation! See Allen-Ferrar [?, p. 480-481].

§43. TRIALITARIAN ALGEBRAS 555

Proof of (??): Let ∆ = ∆(L) be the discriminant algebra of L. The F -algebra

P = L — ∆ — K is a G-Galois algebra where G = S3 — Z/2Z, S3 acts on the Galois

S3 -closure L—∆ and Z/2Z acts on K. We have L—P P —P —P and we may view

L—P as a Galois G-algebra over L. The group S3 acts through permutations of the

factors. Let ιK be a generator of Gal(K/F ) and ρ be a generator of Gal(L — ∆/∆).

Let σ = ρ — ιK , so that σ generates a cyclic subgroup of S3 — Z/2Z of order 6 and

G is generated by σ and 1 — ι∆ — 1. Let (Cs , ns ) be the split Cayley algebra over F

and (C, n) = (Cs , ns ) — P . As in §??, let (x, y) ’ x y = x y be the symmetric

composition on C. The trialitarian structure on E(a) over L is constructed by

Galois descent from the split trialitarian structure End(C) — End(C) — End(C) over

P — P — P:

(43.13) Lemma. Let a ∈ L— be such that NL/F (a) = 1. There exist similitudes

t, t+ , t’ of (C, n) with multipliers a, σ(a), σ 2 (a), respectively, such that:

(1) at(x y) = t+ (x) t’ (y).

(2) t, t+ , t’ ∈ Sym EndP (C), σn , and (t, t+ , t’ )2 = a, σ(a), σ 2 (a) .

(3) σt = t+ σ, σt+ = t’ σ, σt’ = tσ.

(4) One has

(π — ι∆ — 1)t = t(π — ι∆ — 1),

(π — ι∆ — 1)t+ = t’ (π — ι∆ — 1),

(π — ι∆ — 1)t’ = t+ (π — ι∆ — 1),

(π — ι∆ — 1)(1 — σ) = (1 — σ 2 )(π — ι∆ — 1).

Proof : Lemma (??) applied over P to a1 = a, a2 = σ(a), a3 = σ 2 (a) gives (??)

and (??).

(??) and (??) can easily be veri¬ed using the explicit form of t, t+ , and t’

given in Lemma (??).

We now describe the descent de¬ning E(a). Let t, σ and π be the automor-

phisms of C — C — C given by t = (t, t’ , t+ ), σ(x, y, z) = (σy, σz, σx), and

π(x, y, z) = π — ι∆ — 1(x), π — ι∆ — 1(z), π — ι∆ — 1(y) .

It follows from the description of (t, t’ , t+ ) that tσ = σ t, tπ = π t, and σπ = πσ 2 .

Further tσ is σ-linear, tπ is ι-linear and, by (??) of Lemma (??), Int(t) is an

automorphism of the trialitarian algebra End(C)—End(C)—End(C) over P —P —P .

Thus {Int(tσ), Int(tπ)} gives a G-Galois action on End(C) — End(C) — End(C). By

Galois descent we obtain a trialitarian algebra E(a) = (E, L, σ, ±) over L. We claim

that E M4 (K — L/L, a) . Since L/F is cubic, it su¬ces to check that

E—L M4 (K — L/L, a) — E2

for some L — ∆-algebra E2 . Let E — L = E1 — E2 . The (L — L)-algebra E — L is

the descent of End(C) under {Int(tσ)3 , Int(tπ)}. Since [tσ 3 ]2 = a ∈ L, we have

[E — L] = [(K — L/L, a)], [E2 ] ∈ Br(L — L — ∆),

hence the claim.

For ¬xed extensions K and L over F , the trialitarian algebras E(a) are classi¬ed

by L— /NK—L/L (K — L)— :

556 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

(43.14) Proposition. The following conditions are equivalent:

(1) E(a1 ) E(a2 ) as trialitarian algebras.

(2) E(a1 ) E(a2 ) as L-algebras (without involutions).

(3) a1 a’1 ∈ NK—L/L (K — L)— .

2

Proof : (??) implies (??) and the equivalence of (??) and (??) is classical for cyclic

algebras, see for example Corollary (??).

We show that (??) implies (??) following [?]. Assume that a1 = a2 · »ιK (»)

for » ∈ L — K. We have »ιK (») = »σ 3 (») ∈ P — . It follows from NL/F (a1 ) =

1 = NL/F (a2 ) that NP/F (») = 1, so that, by choosing µ = a2 σ 4 (»)σ 5 (») ’1 ,

we deduce µσ 2 (µ)σ 4 (µ) = 1. Now let t = (t1 , t2 , t3 ) be given by Lemma (??) for

a1 = µ, a2 = σ 2 (µ), and a3 = σ 4 (µ). Let c(ai ) be the map tσ as used in the descent

de¬ning E(a) for a = ai , i = 1, 2. A straightforward computation shows that

T ’1 c(a1 )T = c(a2 ) σ(»)σ 2 (»), σ 3 (»)σ 4 (»), σ 5 (»)» .

This implies (by descent) that E(a1 ) E(a2 ).

43.C. Trialitarian algebras of type 2D4 . We say that a trialitarian algebra

T = (E, L, ∆, σ, ±i ) of type 1D4 if L is split, 2D4 if L = F — K for K a quadratic

separable ¬eld extension over F isomorphic to ∆, 3D4 if L is a cyclic ¬eld extension

of F and 6D4 if L — ∆ is a Galois ¬eld extension with group S3 over F .

We now describe trialitarian algebras over an algebra L = F — ∆ where ∆

is quadratic (and is the discriminant algebra of L), i.e., is of type 1D4 or 2D4 .

The results of this section were obtained in collaboration with R. Parimala and R.

Sridharan.

(43.15) Proposition. Let (A, σ) be a central simple F -algebra of degree 8 with an

orthogonal involution and let Z be the center of C(A, σ).

(1) The central simple algebra with involution (A, σ) — C(A, σ), σ over F — Z

admits the structure of a trialitarian algebra T (A, σ) and is functorial in (A, σ).

(2) If T = (A — B, F — ∆, σA — σB , ±) is a trialitarian algebra over L = F — ∆

for ∆ a quadratic ´tale F -algebra, then there exists, after ¬xing a generator ρ of

e

∼

Gal(L — ∆/∆), a unique isomorphism φ : T ’ T (A, σ) of trialitarian algebras such

’

that φ|A = 1|A .

∼

Proof : (??) Let ι be the conjugation on Z. The isomorphism Z — Z ’ Z — Z

’

given by x — y ’ xy, xι(y) induces an isomorphism

∼

±1 : C(A — Z) ’ C(A, σ) — ι C(A, σ).

’

Thus A—Z, C(A, σ), ι C(A, σ) is a trialitarian triple over Z. By triality ±1 induces

a Z-isomorphism

+

∼

±2 = θ ±1 : C C(A, σ), σ ’ ι C(A, σ) — A — Z,

’

so that ± = (1, ±2 ) is an (F — Z)-isomorphism

∼

± : C A — C(A, σ) ’ C(A, σ) — ι C(A, σ) — A — Z.

’

On the other hand we have

∼

C(A, σ) — ι C(A, σ) — A — Z ’ ρ A — Z — C(A, σ) — ι C(A, σ)

’

∼

’ ρ A — C(A, σ) — Z

’

§43. TRIALITARIAN ALGEBRAS 557

for ρ ∈ AutZ (Z — Z — Z) = Gal (F — Z) — Z/Z given by ρ(z0 , z1 , z2 ) = (z1 , z2 , z0 ).

Thus ± can be viewed as an isomorphism

∼

± : C A — C(A, σ) ’ ρ A — C(A, σ) — Z .

’

It is easy to check that ± is trialitarian by splitting Z.

(??) Let

∼

β : C(A — B) ’ ρ (A — B) — Z

’

be a trialitarian structure for (A — B, σA — σB ). Then β is an L — Z isomorphism

∼

β : C(A) — C(B) ’ B — ι B — A — Z

’

∼

and splits as (β1 , β2 ) where β1 : C(A) ’ B and β2 is determined by β1 through

’

triality. Then

∼

β = (1, β1 ) : A — C(A) ’ A — B

’

is an isomorphism of T (A, σ) with (A — B, σA — σB , β). This follows from the

fact that a trialitarian algebra over a product F — Z is determined by the ¬rst

component.

(43.16) Corollary. Let (E, σ) be such that there exists an isomorphism

∼

± : C(E, σ) ’ ρ (E — ∆)

’

(not necessarily trialitarian). If L is not a ¬eld, then there exists a trialitarian

∼

isomorphism ±E : C(E, σ) ’ ρ (E — ∆).

’

Proof : Let E = A — B and write L = F — K = Z(A) — Z(B). Then ± = (±1 , ±2 )

∼ ∼

with ±1 : C(A, σA ) ’ (B, σB ) and ±2 : C(B, σB ) ’ ιB — A — K. On the other

’ ’

hand

∼

±1 — 1K : C(A — K, σ — 1) ’ (B, σB ) — K = (B, σB ) — ι (B, σB )

’

induces by triality an isomorphism

∼

±2 : C(B, σB ) ’ ιB — A — K.

’

The pair ±E = (±1 , ±2 ) is trialitarian.

(43.17) Corollary. Let A, A be central simple F -algebras of degree 8 with or-

thogonal involutions σ, σ and let Z, Z be the centers of C(A, σ), resp. C(A , σ ).

Then the F -algebras C(A, σ), σ and C(A , σ ), σ are isomorphic (as algebras

with involution) if and only if (A, σ) — Z and (A , σ ) — Z are isomorphic (as

F -algebras).

∼

Proof : Any isomorphism φ : C(A, σ) ’ C(A , σ ) induces an isomorphism

’

∼ ∼ ∼

ι

C(A, σ) — A — Z ’ C C(A, σ), σ ’ C C(A , σ ), σ ’ ι C(A , σ ) — A — Z .

’ ’ ’

Looking at all possible components of C(φ) and taking in account that by as-

∼ ∼

sumption C(A, σ) ’ C(A , σ ) gives an isomorphism (A, σ) — Z ’ (A , σ ) — Z .

’ ’

∼

Conversely, any isomorphism (A, σ) — Z ’ (A , σ ) — Z induces an isomorphism

’

∼ ∼

C(A, σ) — Z ’ C(A , σ ) — Z . Since C(A, σ) — Z ’ C(A, σ) —ι C(A, σ), compos-

’ ’

ing with the inclusion C(A, σ) ’ C(A, σ) — Z and the projection C(A , σ ) — Z ’

C(A , σ ) gives a homomorphism C(A, σ) ’ C(A , σ ) of algebras with involution.

This must be an isomorphism since C(A, σ) is central simple over Z.

558 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

(43.18) Corollary ([?]). Let (V, q) and (V , q ) be quadratic spaces of rank 8 and

let Z, Z be the centers of C(V, q), resp. C(V , q ). Then C0 (V, q) and C0 (V , q )

are isomorphic (as algebras over F with involution) if and only if (V, q) — Z and

(V , q ) — Z are similar.

∼

Proof : Since any isomorphism EndF (V ), σq ’ EndF (V ), σq is induced by a

’

∼

similitude (V, q) ’ (V , q ) and vice versa the result follows from Corollary (??).

’

§44. Classi¬cation of Algebras and Groups of Type D4

Let (C, n) be a Cayley algebra with norm n over F , let C = C — (F — F — F ) be

the induced twisted composition and let End(C) be the induced trialitarian algebra

(see Example (??)). Since S3 acts by triality on PGO(C, n), we have a split exact

sequence

p

(44.1) 1 ’ PGO(C, n) ’ PGO(C, n) S 3 ’ S3 ’ 1

’

where p is the projection.

(44.2) Proposition. We have

AutF End(C) PGO+ (C, n)(F ) S3 .

Proof : One shows as in the proof of Proposition (??) that the restriction map

ρ : AutF End(C) ’ AutF (F — F — F ) = S3

has a section. Thus it su¬ces to check that ker ρ = PGO+ (C, n)(F ). Any β in ker ρ

is of the form Int(t) where t = (t0 , t1 , t2 ) is a (F — F — F )-similitude of C — C — C

with multiplier » = (»0 , »1 , »2 ), such that

±C —¦ C0 (t) = Int(t) — 1 —¦ ±C .

It follows from the explicit description of ±C given in (??) that

»’1 t(x) — z — t(y) = t x — t’1 (z) — y

(44.3)

for all x, y, z ∈ C, where x — y = (¯1 y2 , x2 y0 , x0 y1 ) for x = (x0 , x1 , x2 ), y =

x¯ ¯¯ ¯¯

(y0 , y1 , y2 ), multiplication is in the Cayley algebra and x ’ x is conjugation. Con-

¯

dition (??) gives three relations for (t0 , t1 , t2 ):

ti xi+1 (yi+1 zi ) = »’1 ti+1 (xi+1 ) ti+1 (yi+1 )ti (zi ) ,

(44.4) ¯ i = 0, 1, 2.

i

We claim that the group homomorphism

Int(t) ∈ ker ρ ’ [t0 ] ∈ PGO+ (C, n)(F )

is an isomorphism. It is surjective since, by triality, there exist t1 = (t0 )’ , t2 = (t0 )+

such that t = (t0 , t1 , t2 ) (see Proposition (??)). We check that it is injective: let

[t0 ] = 1, so that t0 = µ0 · 1C for some µ0 ∈ F — . It follows from Equation (??)