t2 x(yz) = »’1 µ2 x yt2 (z)

0

2

holds for all x, y, z ∈ C. By putting y = z = 1 we obtain t2 (x) = »’1 µ2 xt2 (1).

0

2

This implies, with a = t2 (1), that x(yz) a = x x(za) . Hence a = t2 (1) is central

in C and the class of t2 in PGO+ (C, n)(F ) is trivial. One shows similarly that the

class of t1 is trivial and, as claimed, ker ρ PGO+ (C, n)(F ).

§44. CLASSIFICATION OF ALGEBRAS AND GROUPS OF TYPE D4 559

(44.5) Corollary. The pointed set H 1 (F, PGO+ S3 ) classi¬es trialitarian F -

8

algebras up to isomorphism. In the exact sequence

H 1 (F, PGO+ ) ’ H 1 (F, PGO+ S3 ) ’ H 1 (F, S3 )

8 8

induced by the exact sequence (??), the ¬rst map associates the trialitarian algebra

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

where ±(A,B,C) is determined as in Proposition (??), to the triple (A, B, C). The

second map associates the class of L to the trialitarian algebra T = (E, L, σ, ±).

Proof : Over a separable closure of F , L and E split, hence the trialitarian algebra

is isomorphic to a split trialitarian algebra Ts . We let it as an exercise to identify

Aut(Ts ) with AutG (w) for some tensor w ∈ W and some representation G ’

GL(W ) such that H 1 (F, G) = 0 (see the proof of Theorem (??)). Then (??)

follows from Proposition (??).

44.A. Groups of trialitarian type D4 . Let T = (E, L, σ, ±) be a trialitarian

F -algebra. The group scheme AutL (T ) of automorphisms of T which are the

identity on L is the connected component of the identity of AutF (T ). We have,

for R ∈ Alg F ,

AutL (T )(R) = { φ ∈ RL/F PGO+ (E, σ) (R) | ±E —¦ C(φ) = (φ — 1) —¦ ±E }

and we set PGO+ (T ) = AutL (T ). Similarly we set

GO+ (T )(R) =

{ x ∈ RL/F GO+ (E, σ) (R) | ±ER —¦ C Int(x) = Int(x) — 1 —¦ ±ER },

so that PGO+ (T ) = GO+ (T )/ Gm , and

Spin(T )(R) = { x ∈ RL/F Spin(E, σ) (R) | ±ER (x) = χ(x) — 1 }.

(44.6) Lemma. For the split trialitarian algebra Ts we have

PGO+ (Ts ) PGO+ (Cs , ns ).

Spin(Ts ) Spin(Cs , ns ) and

Proof : Let Ts = (E, F — F — F, σ, ±E ) with E = A — A — A, A = EndF (C) the

split trialitarian algebra. For x ∈ Spin(E, σ)(R), we have ±ER (x) = χ(x) — 1 if

and only if x = (t, t1 , t2 ) and t1 (x y) = t(x) t2 (y), hence the claim for Spin(Ts ).

The claim for PGO+ (Ts ) follows along similar lines.

Since Spin(Ts ) Spin(Cs , ns ), Spin(T ) is simply connected of type D4 and

the vector representation induces a homomorphism

χ : Spin(T ) ’ PGO+ (T )

which is a surjection of algebraic group schemes. Thus Spin(T ) is the simply

connected cover of PGO+ (T ). Let γ be a cocycle in H 1 (F, PGO+ S3 ) de¬ning

8

the trialitarian algebra T . Since

PGO+ S3 = Aut(Spin8 ) = Aut(PGO+ ),

8 8

we may use γ to twist the Galois action on Spin8 or PGO+ and we have

8

Spin(T ) and (PGO+ )γ (F ) PGO+ (T ).

(Spin8 )γ (F ) 8

(44.7) Remark. If G is of type 1D4 or 2D4 , i.e., if L = F —Z, then E = A—C(A, σ)

and PGO+ (T ) PGO+ (A), Spin(T ) Spin(A).

560 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

Classi¬cation of simple groups of type D4 . Consider the groupoid D4 =

D4 (F ), of trialitarian F -algebras. Denote by D 4 = D 4 (F ) (resp. D 4 = D 4 (F )) the

groupoid of simply connected (resp. adjoint) simple groups of type D4 over F where

morphisms are group isomorphisms. We have functors

S4 : D4 (F ) ’ D 4 (F ) and S 4 : D4 (F ) ’ D 4 (F )

de¬ned by S4 (T ) = Spin(T ), S 4 (T ) = PGO+ (T ).

(44.8) Theorem. The functors S4 : D4 (F ) ’ D 4 (F ) and S 4 : D4 (F ) ’ D 4 (F )

are equivalences of categories.

Proof : Since the natural functor D 4 (F ) ’ D 4 (F ) is an equivalence by Theorem

(??), it su¬ces to prove that S 4 is an equivalence. Let “ = Gal(Fsep /F ). The

¬eld extension functor j : D4 (F ) ’ D4 (Fsep ) is clearly a “-embedding. We show

¬rst that the functor j satis¬es the descent condition. Let T = (E, L, σ, ±) be some

object in D4 (F ) (split, for example). Consider the F -space

W = HomF (E —F E, E) • HomF (E, E) • HomF C(E, σ), E —F ∆(L) ,

the element w = (m, σ, ±) ∈ W where m is the multiplication in E, and the

representation

ρ : GL(E) — GL C(E, σ) ’ GL(W )

given by

ρ(g, h)(x, y, p) = g(x), g(y), h —¦ p —¦ (g — 1)’1

where g(x) and g(y) is the result of the natural action of GL(E) on the ¬rst and

second summands. By Proposition (??) the “-embedding

i : A(ρsep , w) ’ A(ρsep , w)

satis¬es the descent condition. We have a functor

T = T(F ) : A(ρsep , w) ’ D4 (F )

taking w ∈ A(ρsep , w) to the F -space E with the trialitarian structure de¬ned

by w . A morphism between w and w de¬nes an isomorphism of the corresponding

structures on D. The functor T has an evident “-extension

T = T(Fsep ) : A(ρsep , w) ’ D4 (Fsep ),

which is clearly an equivalence of groupoids. Since the functor i satis¬es the descent

condition, so does the functor j.

For the proof of the theorem it su¬ces by Proposition (??) (and the following

Remark (??)) to show that for some T ∈ Dn (F ) the functor T(F ) for a separably

closed ¬eld F induces a group isomorphism

PGO(T ) = AutD4 (F ) (T ) ’ Aut PGO+ (T ) .

(44.9)

The restriction of this homomorphism to the subgroup PGO+ (T ), which is of in-

dex 6, induces an isomorphism of this subgroup with the group of inner auto-

morphisms Int PGO+ (A, σ, f ) , which is a subgroup in Aut PGO+ (A, σ) also

of index 6 (see Theorem (??)). A straightforward computation shows that the

elements θ, θ+ in PGO’ (A, σ, f ) induce outer automorphisms of PGO+ (A, σ, f )

and (??) is an isomorphism.

§44. CLASSIFICATION OF ALGEBRAS AND GROUPS OF TYPE D4 561

Tits algebras. If (A, σ) is a degree 8 algebra with an orthogonal involution,

the description of the Tits algebra of G = Spin(A, σ) is given in (??). Now let

T = (E, L, σ, ±) be a trialitarian algebra with L is a cubic ¬eld extension and let

G = Spin(T ). The Galois group “ acts on C — through Gal(L — Z/F ). There exists

some χ ∈ C — such that Fχ = L is the ¬eld of de¬nition of χ. Since GL Spin(E, σ)

by Remark (??), we have Aχ = E for the corresponding Tits algebra.

44.B. The Cli¬ord invariant. The exact sequence (??) of group schemes

χ

1 ’ C ’ Spin8 ’ PGO+ ’ 1

’ 8

where C is the center of Spin8 , induces an exact sequence

χ 1

1 ’ C ’ Spin8 S3 ’ ’ PGO+ S3 ’ 1

’’ 8

which leads to an exact sequence in cohomology

1)1

(χ

S3 ) ’ ’ ’ H 1 (F, PGO+ S3 ).

’ H 1 (F, C) ’ H 1 (F, Spin8

(44.10) ’ ’’ 8

Since C is not central in Spin8 S3 , there is no connecting homomorphism from

the pointed set H 1 (F, PGO+ S3 ) to H 2 (F, C). However we can obtain a con-

8

necting homomorphism over a ¬xed cubic extension L0 by “twisting” the action of

Gal(Fsep /F ) on each term of the exact sequence (??) through the cocycle δ : Gal(Fsep /F ) ’

S3 de¬ning L0 . We have a sequence of Galois modules

χ

1 ’ (C)δ ’ (Spin8 )δ ’ δ (PGO+ )δ ’ 1.

(44.11) ’ 8

In turn (??) leads to a sequence in cohomology

1

Sn1

χ

(44.12) H (F, Cδ ) ’ H F, (Spin8 )δ ’ ’ H 1 F, (PGO+ )δ ’ ’ H 2 (F, Cδ ).

1 1 δ

’ ’

8

∼

The set H 1 F, (PGO+ )δ classi¬es pairs (T, φ : L ’ L0 ) where T = (E, L, σ, ±)

’

8

is a trialitarian algebra. Moreover the group AutF (L0 ) acts on the pointed set

H 1 F, (PGO+ )δ and H 1 F, (PGO+ )δ / AutF (L0 ) classi¬es trialitarian algebras

8 8

(E, L, σ, ±) with L L0 .

∼

The map H 1 F, (PGO+ )δ ’ H 1 (F, PGO+ S3 ), [T, φ : L ’ L0 ] ’ [T ] has

’

8 8

(p1 )’1 ([L0 ]) as image where

p1 : H 1 (F, PGO+ S3 ) ’ H 1 (F, S3 )

8

maps the class of a trialitarian algebra (E, L, σ, ±E ) to the class of the cubic ex-

tension L. Corresponding results hold for (Spin8 )δ ; in particular H 1 F, (Spin8 )δ

classi¬es pairs

∼

“ = (V, L, Q, β), φ : L ’ L0

’

where “ = (V, L, Q, β) is a twisted composition. The map

H 1 F, (Spin8 )δ ’ H 1 F, (PGO+ )δ

8

associates to (“, φ) the pair End(“), φ . (See Example (??) for the de¬nition of

End(“).)

We call the class Sn1 ([T, φ]) ∈ H 2 (F, Cδ ) the Cli¬ord invariant of T and denote

it by c(T ). Observe that it depends on the choice of a ¬xed L0 -structure on E.

(44.13) Proposition. If L = F — Z and T = T (A, σ) = (A, σ) — C(A, σ), σ ,

then c(T ) = [C(A, σ)] ∈ Br(Z).

562 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

Proof : The image of the homomorphism δ : Gal(Fsep /F ) ’ S3 is a subgroup of

order 2 and

C(Fsep ) = µ2 — µ2

(see Proposition (??)). Therefore we have H 2 F, Cδ = H 2 (Z, µ2 ) and it follows

from the long exact sequence (??) that c(T ) = [C(A, σ)] in Br(Z).

The exact sequence (??)

m

1 ’ C ’ µ 2 — µ2 — µ2 ’ µ 2 ’ 1

’

was used to de¬ne the action of S3 on C. As above, if L0 is a ¬xed cubic ´tale e

F -algebra and δ : Gal(Fsep /F ) ’ S3 is a cocycle which de¬nes L0 , we may use δ

to twist the action of Gal(Fsep /F ) on the above sequence and consider the induced

sequence in cohomology:

(44.14) Lemma. For i ≥ 1, there exists a commutative diagram

H i (F, Cδ ) ’ ’ ’ H i F, (µ2 — µ2 — µ2 )δ ’ ’ ’ H i (F, µ2 )

’’ ’’

¦ ¦

¦ ¦

i

NL/F

NL—∆/L

i i

’ ’ ’ H i (F, µ2 )

H L — ∆(L), µ2 ’’’

’’’ H (L, µ2 ) ’’

where the ¬rst row is exact, the ¬rst vertical map is injective and the second is an

isomorphism. In particular we have

H i (F, Cδ ) ker[NL/F : H i (L, µ2 ) ’ H i (F, µ2 )].

i

Proof : The ¬rst vertical map is the composition of the restriction homomorphism

H i (F, Cδ ) ’ H i (L, Cδ ),

which is injective since [L : F ] = 3, with the isomorphism

∼

•i : H i (L, Cδ ) = H i L, RL—∆/L(µ2 ) ’ H i (L — ∆, µ2 )

’

(see Lemma (??) and Remark (??)). The map µ2 (L) ’ µ2 (L — Fsep ) yields an

isomorphism

∼

RL/F (µ2 ) ’ (µ2 — µ2 — µ2 )δ

’

so that, by Lemma (??), we have an isomorphism

∼

H i F, (µ2 — µ2 — µ2 )δ ’ H i (L, µ2 ).

’

Commutativity follows from the de¬nition of the corestriction.

By Lemma (??) we have maps

∼

ν1 : H 2 (F, Cδ ) ’ H 2 (L, Cδ ) ’ H 2 L — ∆(L), µ2 ,

’

∼

ν2 : H 2 (F, Cδ ) ’ H 2 F, (µ2 — µ2 — µ2 )δ ’ H 2 (L, µ2 ),

’

(44.15) Proposition. The image of the Cli¬ord invariant c(T ) under ν1 is the

class [C(E, σ)] ∈ Br(L — ∆) and its image under ν2 is the class [E] ∈ Br(L).

Proof : The claim follows from Proposition (??) if L is not a ¬eld and the general

case follows by tensoring with L.

§45. LIE ALGEBRAS AND TRIALITY 563

Twisted compositions and trialitarian algebras. We conclude this section

with a characterization of trialitarian algebras T = (E, L, σ, ±E ) such that [E] =

1 ∈ Br(L).

(44.16) Proposition. (1) If T = (E, L, σ, ±E ) is a trialitarian algebra such that

[E] = 1 ∈ Br(L), then there exists a twisted composition “ = (V, L, N, β) such that

T = End(“).

(2) “, “ are twisted compositions such that End(“) End(“ ) if and only if there

exists » ∈ L— such that “ “» .

Proof : (??) The trialitarian algebra (E, L, σ, ±) is of the form End(“) if and only

1)1 of sequence (??). Thus, in view

if its class is in the image of the map (χ

of (??), the assertion will follow if we can show that the condition [E] = 1 in Br(L)

implies Sn1 ([x]) = 0 for [x] = [T, φ] = [(E, L, σ, ±), φ] ∈ H 1 F, (PGO+ )δ . We ¬rst

8

consider the case where L = F — ∆, so that E = A, C(A) (see Proposition (??)).

The homomorphism δ factors through S2 and the action on C = µ2 — µ2 in the

sequence (??) is the twist. Thus C = µ2 — µ2 is a permutation module. By

Lemma (??) and Remark (??) , we have

H 2 (F, Cδ ) H 2 (∆, µ2 )

and Sn1 ([x]) = [C(A, σ)] (see Proposition (??)). Thus [E] = 1 implies Sn1 ([x]) = 1

as wanted. If L is a ¬eld, we extend scalars from F to L. Since L is a cubic extension,

the restriction map H 2 (F, Cδ ) ’ H 2 (L, Cδ ) is injective and, since L—L L—L—∆,

we are reduced to the case L = F — ∆.

(??) The group H 1 (F, Cδ ) operates transitively on the ¬bers of (χ 1)1 ; recall

that by (??)

H 1 (F, Cδ ) = ker[NL/F : L— /L—2 ’ F — /F —2 ].

1

On the other hand we have an exact sequence

1

NL/F

—2 #

— —2 — — —2

’’ F — /F —2

1 ’ F /F ’ L /L ’ L /L

’ ’’

by Proposition (??), hence H 1 (F, Cδ ) im(#) ‚ L— /L—2 . One can then check

that, for [»# ] ∈ H 1 (F, Cδ ), [»# ] acts on [“, φ] as [»# ] · [“, φ] = [“» , φ].

Now let “, “ be such that End(“) End(“ ). We may assume that “, “ are

de¬ned over the same ´tale algebra L. Furthermore, since the action of Aut F (L)

e

is equivariant with respect to the map (χ )1 of sequence (??), we may assume that

δ

we have pairs (“, φ), (“ , φ ) such that End(“), φ End(“ ), φ . Then (“, φ),

(“ , φ ) are in the same ¬ber and the claim follows from the de¬nition of the action

of H 1 (F, Cδ ) on this ¬ber.

§45. Lie Algebras and Triality

In this section we describe how trialitarian algebras are related to Lie algebras

of type D4 . Most of the proofs will only be sketched. We still assume that char F =

2. We write o8 for the Lie algebra of the orthogonal group O(V, q) where q is a

hyperbolic quadratic form of rank 8. As for the groups Spin8 and PGO+ , there 8

exists an S3 -action on the Lie algebra o8 , which is known as “local triality”. Its

description will again use Cli¬ord algebras. For any quadratic space (V, q) we have

o(V, q) = { f ∈ EndF (V ) | bq (f x, y) + bq (x, f y) = 0 for all x, y ∈ V }.

564 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

It turns out that this Lie algebra can be identi¬ed with a (Lie) subalgebra of the

Cli¬ord algebra C(V, q), as we now show. (Compare Jacobson [?, pp. 231“232].)

(45.1) Lemma. For x, y, z ∈ V we have in C(V, q):

[[x, y], z] = 2 xbq (y, z) ’ ybq (x, z) ∈ V.

Proof : This is a direct computation based on the fact that for v, w ∈ V , bq (v, w) =

vw + wv in C(V, q): For x, y, z ∈ V , we compute:

[[x, y], z] = (xyz + xzy + yzx + zyx)

’ (yxz + yzx + xzy + zxy)

= 2 xbq (y, z) ’ ybq (x, z) ∈ V.

Let [V, V ] ‚ C(V, q) be the subspace spanned by the brackets [x, y] = xy ’ yx

for x, y ∈ V . In view of (??) we may de¬ne a linear map

ad : [V, V ] ’ EndF (V )

by: adξ (z) = [ξ, z] for ξ ∈ [V, V ] and z ∈ V . Lemma (??) yields:

ad[x,y] = 2 x — ˆq (y) ’ y — ˆq (x) for x, y ∈ V .

(45.2) b b

(45.3) Lemma. (1) The following diagram is commutative:

[V, V ] ’’’

’’ C0 (V, q)

¦ ¦

¦ ¦ ·q

ad

EndF (V ) ’ ’ ’ C(EndF (V ), σq )

’’

1

2c

where c is the canonical map and ·q is the canonical identi¬cation of Proposi-

tion (??).

(2) The subspace [V, V ] is a Lie subalgebra of L C0 (V, q) , and ad induces an iso-

morphism of Lie algebras:

∼

ad: [V, V ] ’ o(V, q).

’

(3) The restriction of the canonical map c to o(V, q) yields an injective Lie algebra

homomorphism:

1

2c: o(V, q) ’ L C(End(V ), σq ) .

Proof : (??) follows from (??) and from the de¬nitions of c and ·q .

(??) Jacobi™s identity yields for x, y, u, v ∈ V :

[[u, v], [x, y]] = [[[x, y], v], u] ’ [[[x, y], u], v].

Since Lemma (??) shows that [[x, y], z] ∈ V for all x, y, z ∈ V , it follows that

[[u, v], [x, y]] ∈ [V, V ].

Therefore, [V, V ] is a Lie subalgebra of L C0 (V, q) . Jacobi™s identity also yields:

ad[ξ,ζ] = [adξ , adζ ] for ξ, ζ ∈ [V, V ],

§45. LIE ALGEBRAS AND TRIALITY 565

hence ad is a Lie algebra homomorphism. From (??) it follows for x, y, u, v ∈ V

that:

bq ad[x,y] (u), v = 2(bq (x, v)bq (y, u) ’ bq (y, v)bq (x, u))

= ’bq u, ad[x,y] (v) ,

hence ad[x,y] ∈ o(V, q). Therefore, we may consider ad as a map:

ad : [V, V ] ’ o(V, q).

It only remains to prove that this map is bijective. Let n = dim V . Using an

orthogonal basis of V , it is easily veri¬ed that dim[V, V ] = n(n’1)/2 = dim o(V, q).

On the other hand, since ·q is an isomorphism, (??) shows that ad is injective; it

is therefore also surjective.

(??) Using ·q to identify [V, V ] with a Lie subalgebra of C(End(V ), σq ), we

1

derive from (??) and (??) that the restriction of 2 c to o(V, q) is the inverse of ad.

1

Therefore, 2 c is injective on o(V, q) and is a Lie algebra homomorphism.

We have more in dimension 8:

(45.4) Lemma. Let Z be the center of the even Cli¬ord algebra C0 (q). If V has

dimension 8, the embedding [V, V ] ‚ L C0 (q), „ induces a canonical isomorphism

∼

of Lie Z-algebras [V, V ]—Z ’ L C0 (q), „ . Thus the adjoint representation induces

’

∼

an isomorphism ad : L C0 (q), „ ’ o(q) — Z.

’

Proof : Fixing an orthogonal basis of V , it is easy to check that [V, V ] and Z are

linearly disjoint over F in C0 (q), so that the canonical map is injective. It is

surjective by dimension count.

45.A. Local triality. Let (S, ) be a symmetric composition algebra with

norm n. The following proposition is known as the “triality principle” for the Lie

algebra o(n) or as “local triality”.

(45.5) Proposition. For any » ∈ o(n), there exist unique elements »+ , »’ ∈ o(n)

such that

»+ (x y) = »(x) y + x »’ (y),

(1)

»’ (x y) = »+ (x) y + x »(y),

(2)

»(x y) = »’ (x) y + x »+ (y)

(3)

for all x, y ∈ o(n).

Proof : Let » = adξ |S for ξ ∈ [S, S], so that adξ extends to an inner derivation

∼

of C0 (n), also written adξ . Let ±S : C0 (n) ’ EndF (S) — EndF (S) be as in Propo-

’

sition (??). The derivation ±S —¦ adξ —¦±’1 is equal to ad±(ξ) ; we write ±(ξ) as

S

(»+ , »’ ) and, since adξ commutes with „ , we see that »+ , »’ ∈ o(n). For any

x ∈ S we have

»+ 0 »+ 0

0 0 0

»x x x

= ’

0 »’ 0 »’

r»x 0 rx 0 rx 0

by de¬nition of ±S , or

»+ (x y) ’ x »’ (y) = »(x) y

»’ (y x) ’ »+ (y) x = y »(x).

566 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

This gives formulas (??) and (??).

From (??) we obtain

bn »+ (x y), z = bn »(x) y, z + bn x »’ (y), z .

Since bn (x y, z) = bn (x, y z) and since »’ , » and »+ are in o(n), this implies

’bn x, y »+ (z) = ’bn x, »(y z) + bn x, »’ (y) z

for all x, y, and z in o(n), hence (??). We leave uniqueness as an exercise.

Proposition (??) is a Lie analogue of Proposition (??). We have obvious Lie

analogues of (??) and (??). Let θ + (») = »+ , θ’ (») = »’ .

(45.6) Corollary. For all x, y ∈ o(n) we have

(», »+ , »’ ) ∈ o(n) — o(n) — o(n) »(x y) = »’ (x) y + x »+ (y)

o(n)

and the projections ρ, ρ+ , ρ’ : (», »+ , »’ ) ’ », »+ , »’ give the three irreducible

representations of o(n) of degree 8. The maps θ + , θ’ permute the representa-

tions ρ+ , ρ, ρ’ , hence are outer automorphisms of o(n). They generate a group

isomorphic to A3 and o(n)A3 is the Lie algebra of derivations of the composition

algebra S.

Proof : The projection ρ is the natural representation of o(n) and ρ± correspond

to the half-spin representations. These are the three non-equivalent irreducible

representations of o(n) of degree 8 (see Jacobson [?]). Since θ + , θ’ permute these

representations, they are outer automorphisms.

(45.7) Remark. The Lie algebra of derivations of a symmetric composition S is

a simple Lie algebra of type A2 if S is of type A2 or is of type G2 if S is of type G2 .

If the composition algebra (S, , n) is a para-Cayley algebra (C, , n) with con-

jugation π : x ’ x, we have, as in the case of Spin(C, n), not only an action of A3 ,

¯

but of S3 . For any » ∈ o(n) the element θ(») = π»π belongs to o(n). The auto-

morphisms θ, θ+ and θ’ of o(n) generate a group isomorphic to S3 .

(45.8) Theorem ([?, Theorem 5, p. 26]). The group of F -automorphisms of the

Lie algebra o(n) is isomorphic to the semidirect product PGO+ (n) S3 where

PGO+ operates through inner automorphisms and S3 operates through θ + , θ’

and θ.

Proof : Let • be an automorphism of o(n) and let ρi , i = 1, 2, 3, be the three

irreducible representations of degree 8. Then ρi —¦ • is again an irreducible repre-

sentation of degree 8. By Jacobson [?, Chap. 9], there exist ψ ∈ GL(C) and π ∈ S3

such that

ρi —¦ • = Int(ψ) —¦ ρπ(i) .

By Corollary (??) there exists some π ∈ Aut o(n) such that ρπ(i) = ρi —¦ π. Hence

we obtain

ρi —¦ • = Int(ψ) —¦ ρi —¦ π.

It follows in particular for the natural representation o(n) ’ EndF (C) that • =

Int(ψ) —¦ π. It remains to show that Int(ψ) ∈ PGO+ (n) or that ψ ∈ GO+ (n). For

any x ∈ o(n), we have Int(ψ)(x) ∈ o(n), hence

ˆ’1 (ψxψ ’1 )—ˆn = ˆ’1 ψ — ’1 x— ψ —ˆn = ’ψxψ ’1 = ’ˆ’1 ψ — ’1ˆn xˆ’1 ψ —ˆn ,

b b b b b bb b

n n n n

§45. LIE ALGEBRAS AND TRIALITY 567

so that ψ —ˆn ψˆ’1 is central in EndF (C). Thus there exists some » ∈ F — such that

b bn

ψ —ˆn ψ = »ˆn and ψ is a similitude. The fact that ψ is proper follows from the fact

b b

that Int(ψ) does not switch the two half-spin representations.

A Lie algebra L is of type D4 if L — Fsep o8 . In particular o(n) is of type D4 .

(45.9) Corollary. The pointed set H 1 (F, PGO+ S3 ) classi¬es Lie algebras of

8

type D4 over F .

Proof : If F is separably closed, we have PGO+ (n) = PGO+ , so that Corol-

8

lary (??) follows from Theorem (??) and (??).

45.B. Derivations of twisted compositions. Let “ = (V, L, Q, β) be a

twisted composition and let β(x, y) = β(x + y) ’ β(x) ’ β(y) for x, y ∈ V . An

L-linear map d : V ’ V such that d ∈ o(Q) and

(45.10) d β(x, y) = β(dx, y) + β(x, dy)

is a derivation of “. The set Der(“) = Der(V, L, Q, β) of all derivations of “ is a

Lie algebra under the operation [x, y] = x —¦ y ’ y —¦ x. In fact we have

Der(“) = Lie Spin(V, L, Q, β)

where Spin(V, L, Q, β) is as in §??.

If L/F is cyclic with ρ a generator of Gal(L/F ) and β(x) = x — x, comparing

the ρ-semilinear parts on both sides of (??) shows that (??) is equivalent with

d(x — y) = x — dy + dx — y. If “ = C for C a Cayley algebra, the formula d(x — y) =

x — dy + dx — y and Corollary (??) implies that Der(C) o(n). Hence, by descent,

Der(“) is always a Lie algebra of type D4 .

Let J be an Albert algebra over a ¬eld F of characteristic = 2, 3. The F -

vector space Der(J) is a Lie algebra of type F4 (see Chevalley-Schafer [?] or Schafer

[?, Theorem 4.9, p. 112]). Let L be a cubic ´tale subalgebra of J and let J =

e

J(V, L) = L • V be the corresponding Springer decomposition. Let Der(J/L) be

the F -subspace of Der(J) of derivations which are zero on L. We have an obvious

isomorphism

Der(“) Der(J/L)

obtained by extending any derivation of “ to a derivation of J by mapping L to

zero. Thus Der(J/L) is a Lie algebra of type D4 . Such a Lie algebra is said of

Jordan type. We have thus shown the following:

(45.11) Proposition. Every Lie algebra of Jordan type is isomorphic to Der(“)

for some twisted composition “.

45.C. Lie algebras and trialitarian algebras. We may also associate a Lie

algebra L(T ) to a trialitarian algebra T = (E, L, σ, ±):

L(T ) = { x ∈ L(E, σ) | ±(x) = x — 1 }

where L(E, σ) is the Lie algebra of skew-symmetric elements in (E, σ) and can be

identi¬ed with a Lie subalgebra of C(E, σ) in view of Lemma (??). For T = End(C)

we obtain

L(T ) L EndF (C), σn o(n)

by (??), hence L(T ) is of type D4 . We shall see that any simple Lie algebra of

type D4 is of the form L(T ) for some trialitarian algebra T .

568 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

(45.12) Proposition. The restriction map induces an isomorphism of algebraic

group schemes

∼

Autalg End(C) ’ Autalg o(n) .

’

Proof : The restriction map induces a group homomorphism

AutF End(C) ’ AutF o(n) .

Since o(n) generates C0 (n) over F it generates C0 (n)(F —F —F ) over F — F — F and

the map is injective. To prove surjectivity, we show that any automorphism of o(n)

extends to an automorphism of End(C). The group AutF o(n) is the semidirect

product of the group of inner automorphisms with the group S3 where S3 acts as

in Corollary (??). An inner automorphism is of the form Int(f ) where f is a direct

similitude of (C, n) with multiplier ». By Equation (??) we see that in C(C, n)

»’1 ad[f (x),f (y)] z = 2»’1 f (x)bn f (y), z ’ f (y)bn f (x), z

= f ad[x,y] f ’1 (z) .

Thus

ad —¦C0 (f ) = Int(f ) —¦ ad

holds in the Lie algebra [C, C] ‚ C0 (C, n). Since [C, C] generates C0 (C, n), the

automorphism C0 (f ), Int(f ) of End(C) extends Int(f ). We now extend the auto-

morphisms θ± of o(n) to automorphisms of End(C). Let ν : o(n) ’ C0 (n)(F —F —F ) ,

ξ ’ ξ, ρ1 (ξ), ρ2 (ξ) be the canonical embedding. Since ρ1 ν = νθ+ and ρ2 ν = νθ’ ,

the extension of θ + is (ρ1 , ρ1 ) and the extension of θ ’ is (ρ2 , ρ2 ). Let ρ (x0 , x1 , x2 ) =

(x0 , x2 , x1 ). The fact that ∈ AutF o(n) extends follows from ν = Int(π)ρ ν.

(45.13) Corollary. Any Lie algebra L of type D4 over F is of the form L(T ) for

some trialitarian algebra T which is uniquely determined up to isomorphism by L.

Proof : By (??) trialitarian algebras and Lie algebras of type D4 are classi¬ed by

the same pointed set H 1 (F, PGO+ S3 ) and, in view of (??), the same descent

8

datum associated to a cohomology class gives the trialitarian algebra T and its Lie

subalgebra L(T ).

(45.14) Remark. We denote the trialitarian algebra T = (E, L, σ, ±) correspond-

ing to the Lie algebra L by T (L) = E(L), L(L), σ, ± . The semisimple F -algebra

E(L) (and its center L(L)) was already de¬ned by Jacobson [?] and Allen [?]

through Galois descent for any Lie algebra L of type D4 . More precisely, if L

is a Lie algebra of type D4 , then Ls = L — Fsep can be identi¬ed with

o(ns ) S(ns ) ‚ EndFs (Cs ) — EndFs (Cs ) — EndFs (Cs )

where

(», »+ , »’ ) ∈ o(ns ) — o(ns ) — o(ns ) »(x y) = »’ (x) y + x »+ (y)

S(ns ) =

(see (??)) and E(L) is the associative F -subalgebra of EndFs (Cs ) — EndFs (Cs ) —

EndFs (Cs ) generated by the image of L. The algebra E(L) is called the Allen

invariant of L in Allison [?].

In particular:

NOTES 569

(45.15) Proposition (Jacobson [?, §4]). For (A, σ) a central simple algebra of de-

gree 8 over F with orthogonal involution,

L T (A, σ) L(A, σ)

where T (A, σ) is as in (??). In particular any Lie algebra L of type 1 D4 or 2 D4 is

of the form L(A, σ). The algebra L is of type 1 D4 if and only if the discriminant

of the involution σ is trivial.

We conclude with a result of Allen [?, Theorem I, p. 258]:

(45.16) Proposition (Allen). The Allen invariant of a Lie algebra L of type D 4

is a full matrix ring over its center if and only if the algebra is a Lie algebra of

Jordan type.

Proof : Let L be of type D4 . If [E(L)] = 1 in Br(L) then by Proposition (??)

T (L) End(“) for some twisted composition “. Then L L End(“) , which is

isomorphic to Der(“), and the assertion follows by Proposition (??) Conversely, if

L is of Jordan type, we have L L(T ) for T End(“), “ a twisted composition,

hence the claim.

Exercises

1. Let L/F be a cubic ¬eld extension and let char F = 2. Show that the map

K1 F — K1 L ’ K2 L given by symbols is surjective. Hint: Let L = F (ξ); show

that any » ∈ L is of the form » = (±ξ + β)(γξ + δ)’1 for ±, β, γ, and δ ∈ F .

Thus K2 L is generated by symbols of the form {ξ + β; ’ξ + β }.

2. Describe real and p-adic trialitarian algebras. Reference

3. missing: Add

some more

exercises!

Notes

The notion of a trialitarian algebra de¬ned here seems to be new, and our

de¬nition may be not the ¬nal one. The main reason for assuming characteristic

di¬erent from 2, is that in characteristic 2 we need to work with quadratic pairs.

The involution σ of C(A, σ, f ) is part of a quadratic pair if A has degree 8 (see

∼

the notes of Chapter II). Thus, if C(A, σ, f ) ’ (B, σB ) — (C, σC ) σB and σC will

’

also be parts of quadratic pairs (as it should be by triality!). However we did not

succeed in giving a rational de¬nition of the quadratic pair on C(A, σ, f ).

It may be still useful to explain how we came to the concept of trialitarian

algebras, out of three di¬erent situations:

(I) Having the notion of a twisted composition “ = (V, L, Q, β), which is in

particular a quadratic space (V, Q) over a cubic ´tale algebra L, it is tempting

e

to consider the algebra with involution EndL (V ), σL and to try to describe the

structure induced from the existence of β.

(II) In the study of outer forms of Lie algebras of type D4 Jacobson [?] intro-

duced the semisimple algebra E(L), as de¬ned in Remark (??), and studied the

cases 1D4 and 2D4 ; in particular he proved Proposition (??). The techniques of

Jacobson were then applied by Allen [?] to arbitrary outer forms. Allen proved

570 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

in particular that NL/F E(L) = 1 (see Proposition (??)) and associated a coho-

mological invariant in H 2 (L, Gm ) to the Lie algebra L. In fact this invariant is

just the image in H 2 (L, Gm ) of our Cli¬ord invariant. It is used by Allen in his

proof of Proposition (??). As an application, Allen obtained the classi¬cation of

Lie algebras of type D4 over ¬nite and p-adic ¬elds. In [?] Allison used the algebra

E(L) (which he called the Allen algebra) to construct all Lie algebras of type D4

over a number ¬eld. One step in his proof is Proposition (??) in the special case of

number ¬elds (see [?, Proposition 6.1]).

(III) For any central simple algebra (A, σ) of degree 8 with an orthogonal

involution having trivial discriminant, we have C(A, σ) B — C, with B, C of

degree 8 with an orthogonal involution having trivial discriminant. At this stage

one can easily suspect that triality permutes A, B and C. In connection with (I)

and (II), the next step is to view the triple A, B, C as an algebra over F — F — F ,

and this explains how the Cli¬ord algebra comes into the picture.

Quaternionic trialitarian algebras (see §??) were recently used by Garibaldi

[?] to construct all isotropic algebraic groups of type 3D4 and 6D4 over a ¬eld of

characteristic not 2.