∼

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 (??)

(for i = 2) that

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