o

given by Walde [?] to the construction of exceptional Lie algebras. The construction

was formalized and applied to cubic forms by Petersson and Racine (see for example

[?]).

§??. Tits constructions for ¬elds of characteristic not 2 ¬rst appeared in print

in Jacobson™s book [?], as did the fact, also due to Tits, that any Albert algebra is a

¬rst or second Tits construction. These results were announced by Tits in a talk at

542 IX. CUBIC JORDAN ALGEBRAS

the Oberwolfach meeting “Jordan-Algebren und nicht-assoziativen Algebren, 17“

26.8.1967”. With the kind permission of J. Tits and the Research Institute in

Oberwolfach, we reproduce Tits™ R´sum´:

e e

Exceptional simple Jordan Algebras

(I) Denote by k a ¬eld of characteristic not 2, by A a central simple algebra of

degree 3 over k, by n : A ’ A, tr : A ’ A the reduced norm and reduced trace,

and by — : A—A ’ A the symmetric bilinear product de¬ned by (x—x)x = n(x).

For x ∈ A, set x = 1 (tr(x) ’ x). Let c ∈ k — . In the sum A0 + A1 + A2 of three

2

copies of A, introduce the following product:

x0 y1 z2

1

(xx + x x)0 (x y)1 (zx )2

x0 2

(xy )1 c(y — y )2 (y z)0

y1

1

(z x)2 (yz )0 (z — z )1

z2 c

(II) Denote by a quadratic extension of k, by B a central simple algebra of

degree 3 over , and by σ : B ’ B an involution of the second kind kind such

that k = { x ∈ | xσ = x }. Set B Sym = { x ∈ B | xσ = x }. Let b ∈ B Sym and

c ∈ l— be such that n(b) = cσ c. In the sum B Sym + B— of B Sym and a copy B—

of B, de¬ne a product by

x y

1

(xx + x x) (x y)—

x 2

σ

+ y by σ )+ cσ (y σ — y )b’1

σ

(xy )— (yby

y —

Theorem 1. The 27-dimensional algebras described under (I) and (II) are ex-

ceptional simple Jordan algebras over k. Every such algebra is thus obtained.

Theorem 2. The algebra (I) is split if c ∈ n(A) and division otherwise. The

algebra (II) is reduced if c ∈ n(B) and division otherwise.

Theorem 3. There exists an algebra of type (II) which does not split over any

cyclic extension of degree 2 or 3 of k. (Notice that such an algebra is necessarily

division and is not of type (I)).

(For more details, cf. N. Jacobson. Jordan algebras, a forthcoming book).

J. Tits

Observe that the —-product used by Tits is our —-product divided by 2. The

extension of Tits constructions to cubic structures was carried out by McCrimmon

[?]. Tits constructions were systematically used by Petersson and Racine, see for

example [?] and [?]. Petersson and Racine showed in particular that (with a few

exceptions) simple cubic Jordan structures can be constructed by iteration of the

Tits process ([?], Theorem 3.1). The result can be viewed as a cubic analog to the

theorem of Hurwitz, proved by iterating the Cayley-Dickson process.

Tits constructions can be used to give simple examples of exceptional division

Jordan algebras of dimension 27. The ¬rst examples of such division algebras were

constructed by Albert [?]. They were signi¬cantly more complicated than those

through Tits constructions. Assertions (??) and (??) of Theorem (??) and (??) are

NOTES 543

due to Tits. The nice cohomological proof given here is due to Ferrar and Petersson

[?] (for ¬rst Tits constructions).

§??. The existence of the invariants f3 and f5 was ¬rst noticed by Serre (see

for example [?]). The direct computation of the trace form given here, as well as

Propositions (??) and (??) are due to Rost. Serre suggested the existence of the

invariant g3 . Its de¬nition is due to Rost [?]. An elementary approach to that

invariant can be found in Petersson-Racine [?] and a description in characteristic 3

can be found in Petersson-Racine [?].

544 IX. CUBIC JORDAN ALGEBRAS

CHAPTER X

Trialitarian Central Simple Algebras

We assume in this chapter that F is a ¬eld of characteristic not 2. Triality

for PGO+ , i.e., the action of S3 on PGO+ and its consequences, is the subject of

8 8

this chapter. In the ¬rst section we describe the induced action on H 1 (F, PGO+ ).8

This cohomology set classi¬es ordered triples (A, B, C) of central simple algebras of

∼

degree 8 with involutions of orthogonal type such that C(A, σA ), σ ’ (B, σB ) —

’

(C, σC ). Triality implies that this property is symmetric in A, B and C, and the

induced action of S3 on H 1 (F, PGO+ ) permutes A, B, and C. As an application we

8

give a criterion for an orthogonal involution on an algebra of degree 8 to decompose

as a tensor product of three involutions.

We may view a triple (A, B, C) as above as an algebra over the split ´tale e

algebra F — F — F with orthogonal involution (σA , σB , σC ) and some additional

∼

structure (the fact that C(A, σA ), σ ’ (B, σB ) — (C, σC )). Forms of such “alge-

’

bras”, called trialitarian algebras, are classi¬ed by H 1 (F, PGO+ S3 ). Trialitarian

8

algebras are central simple algebras with orthogonal involution of degree 8 over cu-

bic ´tale F -algebras with a condition relating the central simple algebra and its

e

Cli¬ord algebra. Connected components of automorphism groups of such trialitar-

ian algebras give the outer forms of simple adjoint groups of type D4 .

Trialitarian algebras also occur in the construction of Lie algebras of type D 4 .

Some indications in this direction are in the last section.

§42. Algebras of Degree 8

42.A. Trialitarian triples. The pointed set H 1 (F, PGO8 ) classi¬es central

simple algebras of degree 8 over F with an involution of orthogonal type and

the image of the pointed set H 1 (F, PGO+ ) in H 1 (F, PGO8 ) classi¬es central

8

simple algebras of degree 8 over F with an involution of orthogonal type hav-

ing trivial discriminant (see Remark (??)). More precisely, each cocycle x in

H 1 (F, PGO+ ) determines a central simple F -algebra A(x) of degree 8 with an

8

orthogonal involution σA(x) having trivial discriminant, together with a designa-

tion of the two components C + A(x), σA(x) and C ’ A(x), σA(x) of the Cli¬ord

algebra C A(x), σA(x) . Thus, putting B(x), σB(x) = C + A(x), σA(x) , σ and

C(x), σC(x) = C ’ A(x), σA(x) , σ , x determines an ordered triple

A(x), σA(x) , B(x), σB(x) , C(x), σC(x)

of central simple F -algebras of degree 8 with orthogonal involution. The two com-

ponents of the Cli¬ord algebra C A(x), σA(x) are determined by a nontrivial cen-

tral idempotent e, say B(x) = C A(x), σA(x) e and C(x) = C A(x), σA(x) (1 ’

e). Thus two triples [(A, σA ), (B, σB ), (C, σC )] and [(A , σA ), (B , σB ), (C , σC )],

where B = C(A, σA )e and C = C(A, σA )(1 ’ e), resp. B = C(A , σA) )e and

545

546 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

C = C(A , σA )(1 ’ e ), determine the same class in H 1 (F, PGO+ ) if there exists

8

an isomorphism φ : (A, σA ) ’ (A , σA ) such that C(φ)(e) = e . Now let (A, B, C)

be an ordered triple of central simple algebras of degree 8 with orthogonal involution

such that there exists an isomorphism

∼

±A : C(A, σA ), σ ’ (B, σB ) — (C, σC ).

’

The element e = ±’1 (0, 1) is a central idempotent of (A, σA ), hence determines

A

a designation of the two components of C(A, σA ). Moreover this designation is

independent of the particular choice of ±A , since it depends only on the ordering of

the triple. We call two triples (A, B, C) and (A , B , C ) isomorphic if there exist

isomorphisms of algebras with involution

∼ ∼ ∼

(φ1 : A ’ A , φ2 : B ’ B , φ3 : C ’ C )

’ ’ ’

and isomorphisms ±A , resp. ±A as above, such that

±A —¦ C(φ1 ) = (φ2 , φ3 ) —¦ ±A .

Thus:

(42.1) Lemma. The set H 1 (F, PGO+ ) classi¬es isomorphism classes of ordered

8

triples (A, B, C) of central simple F algebras of degree 8 with involutions of orthog-

onal type and trivial discriminant.

Observe that the ordered triples (A, B, C) and (A, C, B) determine in general

di¬erent objects in H 1 (F, PGO+ ) since they correspond to di¬erent designations

8

of the components of C(A, σA ). In fact the action of S2 on H 1 (F, PGO+ ) induced

8

by the exact sequence of group schemes

d

1 ’ PGO+ (A, σ) ’ PGO(A, σ) ’ S2 ’ 1

’

permutes the classes of (A, B, C) and (A, C, B).

(42.2) Example. Let A1 = EndF (C) and σ1 = σn where C is a split Cayley

algebra with norm n. In view of proposition (??) we have a canonical isomorphism

±C : C(A1 , σ1 ), σ ’ (A2 , σ2 ) — (A3 , σ3 )

where (A2 , σ2 ) and (A3 , σ3 ) are copies of the split algebra (A1 , σ1 ). Thus the ordered

triple (A1 , A2 , A3 ) determines a class in H 1 (F, PGO+ ). Since n is hyperbolic, it

8

corresponds to the trivial class.

The group S3 acts as outer automorphisms on the group scheme PGO+ (see

8

+

1

Proposition (??)). It follows that S3 acts on H (F, PGO8 ).

(42.3) Proposition. The action of S3 on H 1 (F, PGO+ ) induced by the action

8

+

of S3 on PGO8 is by permutations on the triples (A, B, C). More precisely, the

choice of an isomorphism

∼

±A : C(A, σA ), σ ’ (B, σB ) — (C, σC )

’

determines isomorphisms

∼

±B : C(B, σB ), σ ’ (C, σC ) — (A, σA ),

’

∼

±C : C(C, σC ), σ ’ (A, σA ) — (B, σB ).

’

Moreover any one of the three ±A , ±B or ±C determines the two others.

§42. ALGEBRAS OF DEGREE 8 547

Proof : We have PGO+ (Fsep ) = PGO+ (Cs , ns )(Fsep ) and we can use the de-

8

scription of the action of S3 on PGO+ (Cs , ns )(Fsep ) given in Proposition (??).

Let θ and θ± be the automorphisms of PGO+ (Cs , ns ) as de¬ned in (??). Let

x = (γg )g∈Gal(Fsep /F ) with γg = [tg ], tg ∈ O+ (Fsep ), be a cocycle in H 1 (F, PGO+ )

8 8

+

+’ +’

which de¬nes (A, σA ). By de¬nition of (θ , θ ) the map (θ , θ ) : PGO ’

PGO+ — PGO+ factors through Autalg C0 (n), σ — Autalg C0 (n), σ . Hence

+ ’

the cocycle θ + x = θ+ ([tg ]) de¬nes the triple (B, σB , ±B ) and θ’ x = θ’ ([tg ]) de-

¬nes (C, σC , ±C ). The last assertion follows by triality.

(42.4) Example. In the situation of Example (??), where A1 = A2 = A3 =

EndF (C) and ±A1 = ±C we obtain ±A2 = ±A3 = ±C since the trivial cocycle

represents the triple (A1 , A2 , A3 ).

We call an ordered triple (A, B, C) of central simple algebras of degree 8 such

∼

that there exists an isomorphism ±A : C(A, σA ), σ ’ (B, σB ) — (C, σC ) a triali-

’

tarian triple. For any φ ∈ S3 , we write the map ± induced from ±A by the action

+

of S3 as ±φ . For example we have ±θ = ±B .

A

A

(42.5) Proposition. Let (A, B, C) be a trialitarian triple. Triality induces iso-

morphisms

Spin(A, σA ) Spin(B, σB ) Spin(C, σC ),

PGO+ (A, σA ) PGO+ (B, σB ) PGO+ (C, σC ).

Proof : Let γ = γg = [tg ], tg ∈ GO+ (Fsep ), be a 1-cocycle de¬ning (A, σA , ±A ) so

8

that γ + = θ+ γ de¬nes (B, σB , ±B ). Since Int PGO+ (Fsep ) = PGO+ (Fsep ) we

8 8

+

may use γ to twist the Galois action on PGO8 . The isomorphism

θ+ : PGO+ ’ (PGO+ )γ

8 8

then is a Galois equivariant map, which descends to an isomorphism

∼

PGO+ (A, σA ) ’ PGO+ (B, σB ).

’

The existence of an isomorphism between corresponding simply connected groups

then follows from Theorem (??).

(42.6) Remark. If (A, σ) is central simple of degree 2n with an orthogonal invo-

lution, the space

L(A, σ) = { x ∈ A | σ(x) = ’x }

of skew-symmetric elements is a Lie algebra of type Dn under the product [x, y] =

xy ’ yx (since it is true over a separable closure of F , see [?, Theorem 9, p. 302]).

In fact L(A, σ) is the Lie algebra of the groups Spin(A, σ) or PGO+ (A, σ) (see

??), so that Proposition (??) implies that

L(A, σA ) L(B, σB ) L(C, σC )

∼

if A is of degree 8 and C(A, σA ) ’ (B, σB ) — (C, σC ). An explicit example where

’

(A, σA ) (B, σB ), but L(A, σA ) L(B, σB ) is in Jacobson [?, Exercise 3, p. 316].

(42.7) Proposition. Let (A, B, C) be a trialitarian triple. We have

(1) [A][B][C] = 1 in Br(F ).

(2) A EndF (V ) if and only if B C.

548 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

(3) A B C if and only if (A, σA ) EndF (C), σn for some Cayley algebra C

with norm n.

Proof : (??) is a special case of Theorem (??), see also Example (??), and (??) is

an immediate consequence of (??).

The “if” direction of (??) is a special case of Proposition (??). For the converse,

it follows from [A] = [B] = [C] = 1 in Br(F ) that (A, σA ) = EndF (V ), σq

and that (V, q) has trivial discriminant and trivial Cli¬ord invariant. In view of

Proposition (??) (V, q) is similar to the norm n of a Cayley algebra C over F . This

implies (A, σA ) EndF (C), σn .

(42.8) Remark. As observed by A. Wadsworth [?], there exist examples of tri-

alitarian triples (A, B, C) such that all algebras A, B, C are division algebras:

Since there exist trialitarian triples EndF (V ), B, B with B a division algebra (see

Dherte [?], Tao [?], or Yanchevski˜ [?]), taking B to be generic with an involution

±

of orthogonal type and trivial discriminant (see Saltman [?]) will give such triples.

42.B. Decomposable involutions. We consider central simple F -algebras

of degree 8 with involutions of orthogonal type which decompose as a tensor prod-

uct of three involutions. In view of Proposition (??) such involutions have trivial

discriminant.

(42.9) Proposition. Let A be a central simple F -algebra of degree 8 with an in-

volution σ of orthogonal type. Then (A, σ) (A1 , σ1 ) — (A2 , σ2 ) — (A3 , σ3 ) with Ai ,

i = 1, 2, 3, quaternion algebras and σi an involution of the ¬rst kind on Ai , if and

only if (A, σ) is isomorphic to C(q0 ), „ where C(q0 ) is the Cli¬ord algebra of a

quadratic space (V0 , q0 ) of rank 6 and „ is the involution which is ’Id on V0 .

Proof : We ¬rst check that the Cli¬ord algebra C(q0 ) admits such a decomposition.

Let q0 = q4 ⊥ q2 be an orthogonal decomposition of q0 with q4 of rank 4 and q2 of

rank 2. Accordingly, we have a decomposition C(q0 ) = C(q4 ) — C(q2 ) where — is

the Z/2Z-graded tensor product (see for example Scharlau [?, p. 328]). Let z be a

generator of the center of C0 (q4 ) such that z 2 = δ4 ∈ F — . The map

φ(x — 1 + 1 — y) = x — 1 + z — y

induces an isomorphism

∼

C(q0 ) = C(q4 ) — C(q2 ) ’ C(δ4 q4 ) — C(q2 )

’

by the universal property of the Cli¬ord algebra. The canonical involution of C(q 0 )

is transported by φ to the tensor product of the two canonical involutions, since z

is invariant by the canonical involution of C(q4 ). Similarly, we may decompose q4

as q4 = q ⊥ q and write

∼

C(q4 ) = C(q ) — C(q ) ’ C(q ) — C(δ q )

’

where δ is the discriminant of q . In this case the canonical involution of C(q )

2

maps a generator z of the center of C0 (q ) such that z = δ ∈ F — to ’z . We

then have to replace the canonical involution of C(q ) (which is of orthogonal type)

by the “second” involution of C(q ), i.e., the involution „ such that „ (x) = ’x

on V . This involution is of symplectic type. Conversely, let

(A, σ) (A1 , σ1 ) — (A2 , σ2 ) — (A3 , σ3 ).

Renumbering the algebras if necessary, we may assume that σ1 is of orthogonal type

and that there exists a quadratic space (V1 , q1 ) such that (A1 , σ1 ) C(q1 ), „1 with

§42. ALGEBRAS OF DEGREE 8 549

„1 the canonical involution of C(q1 ). We may next assume that σ2 and σ3 are of

symplectic type: if σ2 and σ3 are both of orthogonal type, σ2 — σ3 is of orthogonal

type and has trivial discriminant by Proposition (??). Corollary (??) implies that

(A2 , σ2 ) — (A3 , σ3 ) (B, σB ) — (C, σC )

where σB , σC are the canonical involutions of the quaternion algebras B, C, and

we replace (A2 , σ2 ) by (B, σB ), (A3 , σ3 ) by (C, σC ). Then there exist quadratic

forms q2 , q3 such that (A2 , σ2 ) C(q2 ), „2 and (A3 , σ3 ) C(q3 ), „3 , with „2 ,

„3 “second involutions”, as described above. Let δi be the discriminant of qi and

let q0 = q3 ⊥ δ3 q2 ⊥ δ3 δ2 q1 , then (A, σ) C(q0 ), „ .

Algebras C(q0 ), „ occur as factors in special trialitarian triples:

(42.10) Proposition. A triple EndF (V ), B, B is trialitarian if and only if

(B, σB ) C(V0 , q0 ), „ ,

where „ is the involution of C(q0 ) which is ’Id on V0 , for some quadratic space

(V0 , q0 ) of dimension 6.

Proof : Let (A, σ) = EndF (V ), σq be split of degree 8, so that C(A, σ) = C0 (q),

and assume that q has trivial discriminant. Replacing q by »q for some » ∈ F — ,

if necessary, we may assume that q represents 1. Putting q = 1 ⊥ q1 , we de¬ne

∼

an isomorphism ρ : C(’q1 ) ’ C0 (q) by ρ(x) = xv1 where v1 is a generator of 1 .

’

Since the center Z of C0 (q) splits and since C(’q1 ) Z — C0 (’q1 ), we may

∼

’1

view ρ as an isomorphism C0 (q) ’ C0 (’q1 ) — C0 (’q1 ). The center of C0 (q) is

’

¬xed under the canonical involution of C0 (q) since 8 ≡ 0 mod 4. Thus, with the

canonical involution on all three algebras, ρ’1 is an isomorphism of algebras with

involution and the triple

EndF (V ), C0 (’q1 ), C0 (’q1 )

is a trialitarian triple. Let ’q1 = a ⊥ q2 with q2 of rank 6 and let q0 = ’aq2 , then

C0 (’q1 ) C(q0 ) as algebras with involution where the involution on C(q0 ) is the

“second involution”. Thus the triple EndF (V ), C(q0 ), C(q0 ) is trialitarian.

We now characterize fully decomposable involutions on algebras of degree 8:

(42.11) Theorem. Let A be a central simple F -algebra of degree 8 and σ an in-

volution of orthogonal type on A. The following conditions are equivalent:

(1) (A, σ) (A1 , σ1 ) — (A2 , σ2 ) — (A3 , σ3 ) for some quaternion algebras with invo-

lution (Ai , σi ), i = 1, 2, 3.

(2) The involution σ has trivial discriminant and there exists a trialitarian triple

EndF (V ), A, A .

(3) The involution σ has trivial discriminant and one of the factors of C(A, σ)

splits.

Proof : The algebra (A, σ) decomposes if and only if (A, σ) C(q0 ) by Lemma

(??). Thus the equivalence of (??) and (??) follows from (??).

The equivalence of (??) and (??) follows from the fact that S3 operates through

permutations on trialitarian triples.

(42.12) Remark. (Parimala) It follows from Theorem (??) that the condition

[A][B][C] = 1 ∈ Br(F )

550 X. TRIALITARIAN CENTRAL SIMPLE ALGEBRAS

for a trialitarian triple is necessary but not su¬cient. In fact, there exist a ¬eld F

and a central division algebra B of degree 8 with an orthogonal involution over F

which is not a tensor product of three quaternion algebras (see Amitsur-Rowen-

Tignol [?]). That such an algebra always admits an orthogonal involution with

trivial discriminant follows from Parimala-Sridharan-Suresh [?]. Thus, by Theo-

rem (??) there are no orthogonal involutions on B such that M8 (F ), B, B is a

trialitarian triple.

(42.13) Remark. If (A, σ) is central simple with an orthogonal involution which

is hyperbolic, then disc(σ) is trivial and one of the factors of the Cli¬ord algebra

C(A, σ) splits (see (??)). These conditions are also su¬cient for A to have an

orthogonal hyperbolic involution if A has degree 4 (see Proposition (??)) but they

are not su¬cient if A has degree 8 by Theorem (??).

§43. Trialitarian Algebras

43.A. A de¬nition and some properties. Let L be a cubic ´tale F -algebra.

e

We call an L-algebra D such that D — F A — B — C with A , B , C central

simple over F for every ¬eld extension F /F which splits L a central simple L-

algebra. For example any trialitarian triple (A, B, C) is a central simple L-algebra

with an involution of orthogonal type over the split cubic algebra L = F — F — F .

Conversely, let L be a cubic ´tale F -algebra and let E be a central simple algebra of

e

degree 8 with an involution of orthogonal type over L. We want to give conditions

on E/L such that E de¬nes a trialitarian triple over any extension which splits L.

Such a structure will be called a trialitarian algebra. In view of the decomposition

L — L L — L — ∆ where ∆ is the discriminant algebra of L (see (??)), we obtain

a decomposition

(E, σ) — L (E, σ) — (E2 , σ2 )

and (E2 , σ2 ) is an (L — ∆)-central simple algebra with involution of degree 8 over

L—∆, in particular is an L-algebra through the canonical map L ’ L—∆, ’ —1.

As a ¬rst condition we require the existence of an isomorphism of L-algebras with

involution

∼