’

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