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.