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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.

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