In particular, for every central simple algebra with involution (A , σ ) of de-

gree 5, the functor C induces an isomorphism of groups:

O+ (A , σ ) = PGO(A , σ ) =

∼

AutF (A , σ ) ’ AutF C(A , σ ), σ

’

= PGSp C(A , σ ), σ .

We thus recover the third case (deg A = 5) of (??).

Indices. Let (A, σ) ∈ C2 . In order to relate the index of (A, σ) to the Witt

index of the corresponding 5-dimensional quadratic space Symd(A, σ)0 , sσ , we es-

tablish a one-to-one correspondence between isotropic ideals in (A, σ) and isotropic

vectors in Symd(A, σ)0 .

(15.20) Proposition. (1) For every right ideal I ‚ A of reduced dimension 2,

the intersection I © Symd(A, σ) is a 1-dimensional subspace of Symd(A, σ) which

is isotropic for the quadratic form Nrpσ . This subspace is in Symd(A, σ)0 (and

therefore isotropic for the form sσ ) if σ(I) · I = {0}.

(2) For every nonzero vector x ∈ Symd(A, σ) such that Nrpσ (x) = 0, the right ideal

xA has reduced dimension 2. This ideal is isotropic for σ if x ∈ Symd(A, σ)0 .

24 Although(??) is stated for even-dimensional quadratic spaces, the arguments used in the

proof also apply to odd-dimensional spaces.

§15. EXCEPTIONAL ISOMORPHISMS 219

Proof : It su¬ces to prove the proposition over a scalar extension. We may thus

assume that A is split; let (A, σ) = EndF (W ), σb for some 4-dimensional vector

space W with alternating bilinear form b. The bilinear form b induces the standard

identi¬cation •b : W — W = EndF (W ) under which

σ(x — y) = ’y — x and Trd(x — y) = b(y, x)

for x, y ∈ W (see (??)). According to (??), every right ideal I ‚ EndF (W ) of

reduced dimension 2 has the form

I = HomF (W, U ) = U — W

for some 2-dimensional subspace U ‚ W uniquely determined by I. If (u1 , u2 ) is a

basis of U , every element in I has a unique expression of the form u1 —v1 +u2 —v2 for

some v1 , v2 ∈ V . Such an element is symmetrized under σ if and only if v1 = u2 ±

and v2 = ’u1 ± for some ± ∈ F . Therefore,

I © Symd(A, σ) = (u1 — u2 ’ u2 — u1 ) · F,

showing that I © Symd(A, σ) is 1-dimensional. Since the elements in I are not

invertible, it is clear that Nrpσ (x) = 0 for all x ∈ I © Symd(A, σ).

If σ(I) · I = {0}, then σ(u1 — u2 ’ u2 — u1 ) · (u1 — u2 ’ u2 — u1 ) = 0. We have

σ(u1 — u2 ’ u2 — u1 ) · (u1 — u2 ’ u2 — u1 ) = (u1 — u2 ’ u2 — u1 )2

= (u1 — u2 ’ u2 — u1 )b(u2 , u1 )

and, by (??),

Trpσ (u1 — u2 ’ u2 — u1 ) = TrdA (u1 — u2 ) = b(u2 , u1 ).

Hence the condition σ(I) · I = {0} implies that Trpσ (u1 — u2 ’ u2 — u1 ) = 0. This

completes the proof of (??).

In order to prove (??), we choose a basis of V to identify

(A, σ) = M4 (F ), Int(u) —¦ t

for some alternating matrix u ∈ GL4 (F ). Under this identi¬cation, we have

Symd(A, σ) = u · Alt M4 (F ), t .

Since the rank of every alternating matrix is even, it follows that rdim(xA) = 0,

2, or 4 for every x ∈ Alt(A, σ). If Nrpσ (x) = 0, then x is not invertible, hence

rdim(xA) < 4; on the other hand, if x = 0, then rdim(xA) > 0. Therefore,

rdim(xA) = 2 for every nonzero isotropic vector x in Symd(A, σ), Nrpσ . If x ∈

Symd(A, σ)0 , then x2 = ’ Nrpσ (x), hence

σ(xA) · xA = Ax2 A = {0}

if x is isotropic.

This proposition shows that the maps I ’ I © Symd(A, σ) and xF ’ xA

de¬ne a one-to-one correspondence between right ideals of reduced dimension 2 in A

and 1-dimensional isotropic subspaces in Symd(A, σ), Nrpσ . Moreover, under

this bijection isotropic right ideals I for σ correspond to 1-dimensional isotropic

subspaces in Symd(A, σ)0 , sσ .

If A is split, then σ is adjoint to an alternating bilinear form, hence it is

hyperbolic. In particular, if 1 ∈ ind(A, σ), then ind(A, σ) = {0, 1, 2}. Thus, the

only possibilities for the index of (A, σ) are

{0}, {0, 2} and {0, 1, 2},

220 IV. ALGEBRAS OF DEGREE FOUR

and the last case occurs if and only if A is split.

(15.21) Proposition. Let (V, q, ζ) ∈ B2 and (A, σ) ∈ C2 correspond to each other

under the equivalence B2 ≡ C2 . The index of (A, σ) and the Witt index w(V, q) are

related as follows:

ind(A, σ) = {0} ⇐’ w(V, q) = 0;

ind(A, σ) = {0, 2} ⇐’ w(V, q) = 1;

ind(A, σ) = {0, 1, 2} ⇐’ w(V, q) = 2.

Proof : Proposition (??) shows that 2 ∈ ind(A, σ) if and only if w(V, q) > 0. There-

fore, it su¬ces to show that A splits if and only if w(V, q) = 2. If the latter

condition holds, then (V, q) has an orthogonal decomposition (V, q) H(U ) • uF

for some 4-dimensional hyperbolic space H(U ) and some vector u ∈ V such that

q(u) = 1, hence C0 (V, q) C H(U ) . It follows from (??) that C0 (V, q), hence also

A, is split. Conversely, suppose (A, σ) = EndF (W ), σb for some 4-dimensional

vector space W with alternating form b. As in (??), we identify A with W — W

under •b . If (e1 , e2 , e3 , e4 ) is a symplectic basis of W , the span of e1 — e3 ’ e3 — e1

and e1 — e4 ’ e4 — e1 is a totally isotropic subspace of Symd(A, σ), sσ , hence

w(V, q) = 2.

15.D. A3 ≡ D3 . The equivalence between the groupoid A3 of central simple

algebras of degree 4 with involution of the second kind over a quadratic ´tale ex-

e

tension of F and the groupoid D3 of central simple F -algebras of degree 6 with

quadratic pair is given by the Cli¬ord algebra and the discriminant algebra con-

structions. Let

C : D3 ’ A 3

be the functor which maps (A, σ, f ) ∈ D3 to its Cli¬ord algebra with canonical

involution C(A, σ, f ), σ and let

D : A3 ’ D 3

be the functor which maps (B, „ ) ∈ A3 to the discriminant algebra D(B, „ ) with

its canonical quadratic pair („ , fD ).

As in §??, the proof that these functors de¬ne an equivalence of groupoids is

based on a Lie algebra isomorphism which we now describe.

For (B, „ ) ∈ A3 , recall from (??) the Lie algebra

s(B, „ ) = { b ∈ B | b + „ (b) ∈ F and TrdB (b) = 2 b + „ (b) }

and from (??) the Lie algebra homomorphism

™

»2 : B ’ »2 B.

Endowing B and »2 B with the nonsingular symmetric bilinear forms TB and T»2 B ,

we may consider the adjoint F -linear map

™

(»2 )— : »2 B ’ B,

™

which is explicitly de¬ned by the following property: the image (»2 )— (ξ) of ξ ∈ »2 B

is the unique element of B such that

™ ™

TrdB (»2 )— (ξ)y = Trd»2 B (ξ »2 y) for all y ∈ B.

§15. EXCEPTIONAL ISOMORPHISMS 221

™

(15.22) Proposition. The map (»2 )— restricts to a linear map

»— : D(B, „ ) ’ s(B, „ ),

which factors through the canonical map c : D(B, „ ) ’ C D(B, „ ), „ , fD and in-

duces a Lie algebra isomorphism

∼

»— : c D(B, „ ) ’ s(B, „ ).

’

This Lie algebra isomorphism extends to an isomorphism of (associative) F -alge-

bras with involution

∼

C D(B, „ ), „ , fD , „ ’ (B, „ ).

’

Proof : Let γ be the canonical involution on »2 B. For y ∈ B, Proposition (??)

™ ™

yields γ(»2 y) = TrdB (y) ’ »2 y. Therefore, for ξ ∈ »2 B we have

™ ™

Trd»2 B γ(ξ)»2 y = TrdB (y) Trd»2 B (ξ) ’ Trd»2 B (ξ »2 y).

™ ™

By the de¬nition of (»2 )— , this last equality yields (»2 )— γ(ξ) = Trd»2 B (ξ) ’

™ ™ ™

(»2 )— (ξ). Similarly, (»2 )— „ §2 (ξ) = „ (»2 )— (ξ) for ξ ∈ »2 B. Therefore, if ξ ∈

D(B, „ ), i.e., „ §2 (ξ) = γ(ξ), then

™ ™

„ (»2 )— (ξ) + (»2 )— (ξ) = Trd»2 B (ξ) ∈ F.

™

Since, by the de¬nition of (»2 )— ,

™ ™

TrdB (»2 )— (ξ) = Trd»2 B (ξ »2 1) = 2 Trd»2 B (ξ),

™

it follows that (»2 )— (ξ) ∈ s(B, „ ) for ξ ∈ D(B, „ ), proving the ¬rst part.

To prove the rest, we extend scalars to an algebraic closure of F . We may thus

assume that B = EndF (V ) — EndF (V — ) for some 4-dimensional F -vector space V ,

and the involution „ is given by

„ (g, ht ) = (h, g t ) for g, h ∈ EndF (V ).

We may then identify

2

D(B, „ ), „ , fD = EndF ( V ), σq , fq

where (σq , fq ) is the quadratic pair associated with the canonical quadratic map

2 4 4

q: V’ V of (??). Let e ∈ V be a nonzero element (hence a basis). We

4

V = F , hence to view q as a quadratic form on 2 V . The

use e to identify

2 2 2

∼

standard identi¬cation •q : V— V ’ EndF ( V ) then yields an identi¬ca-

’

tion

2 ∼

·q : C0 ( V, q) ’ C D(B, „ ), „ , fD

’

which preserves the canonical involutions.

2 ∼

We next de¬ne an isomorphism C0 ( V, q) ’ EndF (V ) — EndF (V — ) = B by

’

2 ∼

restriction of an isomorphism C( V, q) ’ EndF (V • V — ).

’

For ξ ∈ 2 V , we de¬ne maps ξ : V ’ V — and rξ : V — ’ V by the following

conditions, where , is the canonical pairing of a vector space and its dual:

ξ§x§y =e· ξ (x), y for x, y ∈ V

for ψ, • ∈ V — .

ψ § •, ξ = ψ, rξ (•)

222 IV. ALGEBRAS OF DEGREE FOUR

The map

Hom(V — , V )

End(V )

2

V ’ EndF (V • V — ) =

i:

Hom(V, V — ) End(V — )

0 rξ

2

which carries ξ ∈ V to induces an isomorphism

ξ0

2 ∼

V, q) ’ EndF (V • V — )

(15.23) i— : C( ’

which restricts to an isomorphism of algebras with involution

2 ∼

V, q), „0 ’ EndF (V ) — EndF (V — ), „ .

i— : C 0 ( ’

To complete the proof, it now su¬ces to show that this isomorphism extends »— ,

in the sense that

2

i— (ξ · ·) = »— (ξ — ·) for ξ, · ∈ V.

In view of the de¬nition of »— , this amounts to proving

™

tr(ξ — · —¦ »2 g) = tr(rξ —¦ · —¦ g) for all g ∈ EndF (V ).

Veri¬cation of this formula is left to the reader.

(15.24) Theorem. The functors D and C de¬ne an equivalence of groupoids

A3 ≡ D 3 .

Moreover, if (B, „ ) ∈ A3 and (A, σ, f ) ∈ D3 correspond to each other under this

equivalence, the center Z(B) of B satis¬es

F disc(A, σ, f ) if char F = 2;

Z(B)

F „˜’1 disc(A, σ, f ) if char F = 2.

Proof : Once the equivalence of groupoids has been established, then the algebra B

of degree 4 corresponding to (A, σ, f ) ∈ D3 is the Cli¬ord algebra C(A, σ, f ), hence

the description of Z(B) follows from the structure theorem for Cli¬ord algebras

(??).

In order to prove the ¬rst part, we show that for (A, σ, f ) ∈ D3 and (B, „ ) ∈ A3

there are canonical isomorphisms

(A, σ, f ) D C(A, σ, f ), σ , σ, fD and (B, „ ) C D(B, „ ), „ , fD , „

which yield natural transformations

D —¦ C ∼ IdD C —¦ D ∼ IdA3 .

and

= =

3

Proposition (??) yields a canonical isomorphism

∼

C D(B, „ ), „ , fD , „ ’ (B, „ ).

’

On the other hand, starting with (A, σ, f ) ∈ D3 , we may also apply (??) to get a

Lie algebra isomorphism

∼

»— : c D C(A, σ, f ), σ ’ s C(A, σ, f ), σ .

’

By (??), one may check that c(A) ‚ s C(A, σ, f ), σ , and dimension count shows

that this inclusion is an equality. Since »— extends to an F -algebra isomorphism,

it is the identity on F . Therefore, by (??) it induces a Lie algebra isomorphism

∼

» : Alt D C(A, σ, f ), σ , σ ’ Alt(A, σ).

’

§15. EXCEPTIONAL ISOMORPHISMS 223

We aim to show that this isomorphism extends to an isomorphism of (associative)

F -algebras with quadratic pair

D C(A, σ, f ), σ , σ, fD ’ (A, σ, f ).

By (??), it su¬ces to prove the property over an algebraic closure of F . We may

thus assume that A is the endomorphism algebra of a hyperbolic space, so

A = EndF H(U ) = EndF (U — • U )

where U is a 3-dimensional vector space, U — is its dual, and (σ, f ) = (σqU , fqU ) is

the quadratic pair associated with the hyperbolic form qU . Recall that qU is de¬ned

by

for • ∈ U — , u ∈ U .

qU (• + u) = •(u) = •, u

The Cli¬ord algebra of (A, σ, f ) may be described as follows:

C(A, σ, f ) = EndF ( U ) — EndF ( U ),

0 1

2 3

where 0 U = F • U and 1 U = U • U (see (??)). The involution σ

on C(A, σ) is of the second kind; it interchanges EndF ( 0 U ) and EndF ( 1 U ).

Therefore, the discriminant algebra of C(A, σ), σ is

2

D C(A, σ), σ = EndF ( U) ,

1

and its quadratic pair (σ, fD ) is associated with the canonical quadratic map

2 4

q: ( U) ’ ( U) F.

1 1

Since dim U = 3, there are canonical isomorphisms

2 2 3 4 3 3

( U) = U •( U — U ) and ( U) = U— U

1 1

given by

(u1 + ξ1 ) § (u2 + ξ2 ) = u1 § u2 + ξ1 — u2 ’ ξ2 — u1

and

(u1 + ξ1 ) § (u2 + ξ2 ) § (u3 + ξ3 ) § (u4 + ξ4 ) =

ξ1 — (u2 § u3 § u4 ) ’ ξ2 — (u1 § u3 § u4 )

+ ξ3 — (u1 § u2 § u4 ) ’ ξ4 — (u1 § u2 § u3 )

3

for u1 , . . . , u4 ∈ U and ξ1 , . . . , ξ4 ∈ U . Under these identi¬cations, the canonical

2 3 3 3

quadratic map q : U • ( U — U) ’ U— U is given by

q(θ + ξ — u) = ξ — (u § θ)

2 3 3

for θ ∈ U, ξ ∈ U and u ∈ U . Picking a nonzero element µ ∈ U , we

3 3

identify U— U with F by means of the basis µ — µ; we may thus regard q

as a quadratic form on 2 U • ( 3 U — U ). The discriminant algebra of C(A, σ, f )

then has the alternate description

2 3

D C(A, σ, f ), σ , σ, fD = EndF U •( U — U ) , σq , f q .

In order to de¬ne an isomorphism of this algebra with (A, σ, f ), it su¬ces, by (??),

to de¬ne a similitude of quadratic spaces

2 3

g : (U — • U, qU ) ’ U •( U — U ), q .

224 IV. ALGEBRAS OF DEGREE FOUR

For • ∈ U — and u ∈ U , we set

g(• + u) = θ + µ — u,

2

where θ ∈ U is such that µ · •, x = θ § x for all x ∈ U . We then have

µ — µ · •, u = µ — (u § θ) = q(θ + µ — u),

hence g is an isometry of quadratic spaces. We claim that the inverse of the induced

isomorphism

2 3

∼

g— : EndF (U — • U ), σq ’ EndF

’ U •( U — U ) , σb

extends ». To prove the claim, we use the identi¬cations

2 3

D C(A, σ, f ), σ = EndF U •( U — U)

2 3 2 3

= U •( U — U) — U •( U — U)

and

A = EndF (U — • U ) = (U — • U ) — (U — • U ).

Since c(A) = (U — • U ) · (U — • U ) ‚ C0 H(U ), qU is spanned by elements of the

form (• + u) · (ψ + v) with •, ψ ∈ U — and u, v ∈ U , it su¬ces to show that

the corresponding element (d• + u ) —¦ (dψ + v ) ∈ EndF ( 0 U ) — EndF ( 1 U ) =

C0 H(U ), qU (under the isomorphism of (??)) is »— g(• + u) — g(ψ + v) . This

amounts to verifying that

™

= tr »2 h —¦ g(• + u) — g(ψ + v)

tr h —¦ (d• + u) —¦ (dψ + v)

for all h ∈ EndF ( U ) — EndF ( U ). Details are left to the reader.

0 1

(15.25) Remark. For (B, „ ) ∈ A3 , the Lie isomorphism »— : c D(B, „ ) ’ s(B, „ )

∼

restricts to a Lie isomorphism c D(B, „ ) 0 ’ Skew(B, „ )0 . The inverse of this

’

™

1

isomorphism is 2 c —¦ »2 if char F = 2. Similarly, for (A, σ, f ) ∈ D3 , the inverse of the

™

Lie isomorphism » used in the proof of the theorem above is »2 —¦ 1 c if char F = 2.

2

(15.26) Corollary. For every central simple algebra A of degree 6 with quadratic

pair (σ, f ), the functor C induces an isomorphism of groups

∼

PGO(A, σ, f ) = AutF (A, σ, f ) ’ AutF C(A, σ, f ), σ

’

which restricts into an isomorphism of groups:

∼

PGO+ (A, σ, f ) ’ AutZ(A,σ,f ) C(A, σ, f ), σ = PGU C(A, σ, f ), σ

’

where Z(A, σ, f ) is the center of C(A, σ, f ).

Proof : The ¬rst isomorphism follows from the fact that C de¬nes an equivalence

of groupoids D3 ’ A3 (see (??)). Under this isomorphism, the proper similitudes

correspond to automorphisms of C(A, σ, f ) which restrict to the identity on the

center Z(A, σ, f ), by (??).

We thus recover the fourth case (deg A = 6) of (??).

§15. EXCEPTIONAL ISOMORPHISMS 225

Cli¬ord groups. Let (A, σ, f ) ∈ D3 and (B, „ ) ∈ A3 . Let K be the center

of (B, „ ), and assume that (A, σ, f ) and (B, „ ) correspond to each other under the

groupoid equivalence A3 ≡ D3 , so that we may identify (B, „ ) = C(A, σ, f ), σ and

(A, σ, f ) = D(B, „ ), „ , fD . Our goal is to relate the Cli¬ord groups of (A, σ, f ) to

groups of similitudes of (B, „ ). We write µσ and µ„ for the multiplier maps for the

involutions σ and „ respectively.

(15.27) Proposition. The extended Cli¬ord group of (A, σ, f ) is the group of

similitudes of (B, „ ), i.e.,

„¦(A, σ, f ) = GU(B, „ ),

and the following diagram commutes:

χ

„¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f )

’’

¦

¦

(15.28)

D

GU(B, „ ) ’ ’ ’

’’ Aut(A, σ, f )

where D is the canonical map of §??. The Cli¬ord group of (A, σ, f ) is

“(A, σ, f ) = SGU(B, „ ) = { g ∈ GU(B, „ ) | NrdB (g) = µ„ (g)2 }

and the following diagram commutes:

χ

“(A, σ, f ) ’ ’ ’ O+ (A, σ, f )

’’

¦

¦

(15.29)

»

SGU(B, „ ) ’ ’ ’ O(A, σ, f )

’’

where » (g) = µ„ (g)’1 »2 g ∈ D(B, „ ) = A for g ∈ SGU(B, „ ). Moreover,

Spin(A, σ, f ) = SU(B, „ ) = { g ∈ GU(B, „ ) | NrdB (g) = µ„ (g) = 1 }.

Proof : Since (??) shows that the canonical map

C : PGO+ (A, σ, f ) ’ AutK (B, „ )

is surjective, it follows from the de¬nition of the extended Cli¬ord group in (??) that

„¦(A, σ, f ) = GU(B, „ ). By the de¬nition of χ , the following diagram commutes:

χ

„¦(A, σ, f ) ’ ’ ’ PGO+ (A, σ, f )

’’

¦

¦

C

Int

GU(B, „ ) ’ ’ ’

’’ AutK (B, „ ).

The commutativity of (??) follows, since the inverse of

∼

C : PGO+ (A, σ, f ) ’ AutK (B, „ )

’

is given by the canonical map D; indeed, the groupoid equivalence A3 ≡ D3 is given

by the Cli¬ord and discriminant algebra constructions.

To identify “(A, σ, f ), it su¬ces to prove that the homomorphism

κ : „¦(A, σ, f ) ’ K — /F —

whose kernel is “(A, σ, f ) (see (??)) coincides with the homomorphism

ν : GU(B, „ ) ’ K — /F —

226 IV. ALGEBRAS OF DEGREE FOUR

whose kernel is SGU(B, „ ) (see (??)). The description of Spin(A, σ, f ) also follows,

since Spin(A, σ, f ) = “(A, σ, f )©U(B, „ ). The following lemma therefore completes

the proof:

(15.30) Lemma. Diagram (??) and the following diagram are commutative:

κ

„¦(A, σ, f ) ’ ’ ’ K — /F —

’’

ν

GU(B, „ ) ’ ’ ’ K — /F — .

’’

Proof : It su¬ces to prove commutativity of the diagrams over a scalar extension.

We may thus assume that the base ¬eld F is algebraically closed.

4

Let V be a 4-dimensional vector space over F . Pick a nonzero element e ∈ V

4 2 4

to identify V = F and view the canonical quadratic map q : V’ V

of (??) as a quadratic form. Since F is algebraically closed, we have (A, σ, f )

2

EndF ( V ), σq , fq where (σq , fq ) is the quadratic pair associated with q. We ¬x

such an isomorphism and use it to identify until the end of the proof

2

(A, σ, f ) = EndF ( V ), σq , fq .

2

V ’ EndF (V • V — ) de¬ned in (??) induces an isomorphism

The map i :

2 ∼

V, q) ’ EndF (V • V — )

i— : C( ’

2

which identi¬es the Cli¬ord algebra B = C(A, σ, f ) = C0 ( V, q) with EndF (V ) —

EndF (V — ). The involution „ is then given by

t t

„ (f1 , f2 ) = (f2 , f1 )

for f1 , f2 ∈ EndF (V ). Therefore,

GU(B, „ ) = { f, ρ(f ’1 )t | ρ ∈ F — , f ∈ GL(V ) }.

For g = f, ρ(f ’1 )t ∈ GU(B, „ ), we consider

2

V ) = A and γ = f 2 , det f (f ’2 )t ∈ GU(B, „ ).

f § f ∈ EndF (

A computation shows that

2 2

f § f ∈ GO+ ( V, q) = GO+ (A, σ, f ), γ ∈ “+ ( V, q) = “(A, σ, f )

and moreover

χ(γ) = µ(f § f )’1 (f § f )2 .

Int(g) = C(f § f ),

Therefore, κ(g) = z · F — where z = 1, ρ2 (det f )’1 ∈ K — = F — — F — is such that

g 2 = z · γ. On the other hand, we have

µ„ (g)’2 NrdB (g) = ρ’2 det f, ρ2 (det f )’1 = zι(z)’1 ,

hence ν(g) = z · F — = κ(g).

It remains only to prove the commutativity of (??). Since ν = κ, we have

“(A, σ, f ) = SGU(B, „ ). Therefore, every element in “(A, σ, f ) has the form

f, ρ(f ’1 )t for some ρ ∈ F — and some f ∈ GL(V ). A computation yields

f ’1

f 0 0 2

= ρ’1 i— f § f (ξ)

· i— (ξ) · for ξ ∈ V,

’1 t ’1 t

0 ρ(f ) 0 ρ f

hence χ f, ρ(f ’1 )t = ρ’1 f § f and (??) commutes.

§15. EXCEPTIONAL ISOMORPHISMS 227

(15.31) Corollary. For (A, σ, f ) ∈ D3 , the group of multipliers of proper simili-

tudes and the group of spinor norms of (A, σ, f ) are given by

G+ (A, σ, f ) = { NK/F (z) | zι(z)’1 = µ„ (g)’2 NrdB (g) for some g ∈ GU(B, „ ) }

and

Sn(A, σ, f ) = { µ„ (g) | g ∈ SGU(B, „ ) }.

Proof : In view of the description of „¦(A, σ, f ) and κ above, it follows from (??)

that µσ PGO+ (A, σ, f ) = NK/F —¦ ν GU(B, „ ) , proving the ¬rst relation. The

second relation follows from the description of “(A, σ, f ) in (??).

The case of trivial discriminant. If K/F is a given ´tale quadratic exten-

e

sion, the functors D and C of (??) relate algebras with involution (B, „ ) ∈ A3 with

center K and algebras with involutions (A, σ, f ) ∈ D3 whose Cli¬ord algebra has

center Z(A, σ, f ) K. In order to make explicit the special case where K = F — F ,

let 1A3 be the full subgroupoid of A3 whose objects are algebras of degree 4 over

F — F with involution of the second kind and let 1 D3 be the full subgroupoid of D3

whose objects are algebras of degree 6 with quadratic pair of trivial discriminant.

Every (B, „ ) ∈ 1A3 is isomorphic to an algebra of the form (E — E op , µ) where E is

a central simple F -algebra of degree 4 and µ is the exchange involution, hence 1A3

is also equivalent to the groupoid of algebras of the form (E — E op , µ). Since

D(E — E op , µ), µ, fD = (»2 E, γ, f )

where (γ, f ) is the canonical quadratic pair on »2 E (see (??) if char F = 2), the

following is a special case of (??):

(15.32) Corollary. The Cli¬ord algebra functor C : 1 D3 ’ 1A3 and the functor

D : 1A3 ’ 1 D3 , which maps E — E op , µ to (»2 E, γ, f ), de¬ne an equivalence of

groupoids

1

A3 ≡ 1 D 3 .

In particular, for every central simple F -algebra E of degree 4,

C(»2 E, γ, f ) (E — E op , µ).

Observe that the maps in 1A3 are isomorphisms of algebras over F , not over

F — F . In particular, E — E op , µ and E op — E, µ are isomorphic in 1A3 , under

the map which interchanges the two factors. Therefore, 1A3 is not equivalent to

the groupoid of central simple F -algebras of degree 4 where the maps are the F -

algebra isomorphisms. There is however a correspondence between isomorphism

classes which we now describe.

For (A, σ, f ) ∈ 1 D3 , the Cli¬ord algebra C(A, σ, f ) decomposes into a direct

product

C(A, σ, f ) = C + (A, σ, f ) — C ’ (A, σ, f )

for some central simple F -algebras C + (A, σ, f ), C ’ (A, σ, f ) of degree 4. The

fundamental relations (??) and (??) show that C + (A, σ, f ) C ’ (A, σ, f )op and

C + (A, σ, f )—2 C ’ (A, σ, f )—2 ∼ A.

If (V, q) is a quadratic space of dimension 6 and trivial discriminant, we also

let C ± (V, q) denote C ± EndF (V ), σq , fq . The algebras C + (V, q) and C ’ (V, q) are

isomorphic central simple F -algebras of degree 4 and exponent 2.

228 IV. ALGEBRAS OF DEGREE FOUR

(15.33) Corollary. Every central simple F -algebra of degree 4 and exponent 2 is

of the form C ± (V, q) for some quadratic space (V, q) of dimension 6 and trivial

discriminant, uniquely determined up to similarity.

Every central simple F -algebra of degree 4 and exponent 4 is of the form

C (A, σ, f ) for some (A, σ, f ) ∈ 1 D3 such that ind A = 2, uniquely determined

±

up to isomorphism.

Proof : For every central simple F -algebra E of degree 4, we have E — E op

C + (»2 E, γ, f ) — C ’ (»2 E, γ, f ) by (??), hence

C ± (»2 E, γ, f ).

E

Since »2 E is Brauer-equivalent to E —2 , it is split if E has exponent 2 and has

C ± (A, σ, f ) for some

index 2 if E has exponent 4, by (??). Moreover, if E

(A, σ, f ) ∈ 1 D3 , then (E — E op , µ) C(A, σ, f ), σ since all involutions on E — E op

are isomorphic to the exchange involution. Therefore, by (??), we have

D(E — E op , µ), µ, fD (»2 E, γ, f ).

(A, σ, f )

To complete the proof, observe that when A = EndF (V ) we have (σ, f ) = (σq , fq )

for some quadratic form q, and the quadratic space (V, q) is determined up to

similarity by the algebra with quadratic pair (A, σ, f ) by (??).

Corollaries (??) and (??), and Proposition (??), can also be specialized to the

case where the discriminant of (A, σ, f ) is trivial. In particular, (??) simpli¬es

remarkably:

(15.34) Corollary. Let (A, σ, f ) ∈ 1 D3 and let C(A, σ, f ) E — E op . The multi-

pliers of similitudes of (A, σ, f ) are given by

G(A, σ, f ) = G+ (A, σ, f ) = F —2 · NrdE (E — )

and the spinor norms of (A, σ, f ) by

Sn(A, σ, f ) = { ρ ∈ F — | ρ2 ∈ NrdE (E — ) }.

Proof : The equality G(A, σ, f ) = G+ (A, σ, f ) follows from the hypothesis that

disc(A, σ, f ) is trivial by (??). Since the canonical involution σ on C(A, σ, f ) is the

exchange involution, we have under the identi¬cation C(A, σ, f ) = E — E op that

x, ρ(x’1 )op ρ ∈ F —, x ∈ E— ,

GU C(A, σ, f ), σ =

and, for g = x, ρ(x’1 )op ,

µ(g)’2 NrdC(A,σ,f ) (g) = ρ’2 NrdE (x), ρ2 NrdE (x)’1 = zι(z)’1

with z = NrdE (x), ρ2 ∈ Z(A, σ, f ) = F — F . Since NZ(A,σ,f )/F (z) = ρ2 NrdE (x),

Corollary (??) yields the equality G+ (A, σ, f ) = F —2 · NrdE (E — ). Finally, we have

by (??):

“(A, σ, f ) = SGU C(A, σ, f ), σ

= { x, ρ(x’1 )op ∈ GU C(A, σ, f ), σ | ρ2 = NrdE (x) },

hence

= { ρ ∈ F — | ρ2 ∈ NrdE (E — ) }.

Sn(A, σ, f ) = µ SGU C(A, σ, f ), σ

§15. EXCEPTIONAL ISOMORPHISMS 229

Examples. In this subsection, we explicitly determine the algebra with invo-

lution (B, „ ) ∈ A3 corresponding to (A, σ, f ) ∈ D3 when the quadratic pair (σ, f ) is

isotropic. Since the correspondence is bijective, our computations also yield infor-

mation on the discriminant algebra of some (B, „ ) ∈ A3 , which will be crucial for

relating the indices of (A, σ, f ) and (B, „ ) in (??) and (??) below.

(15.35) Example. Let (A, σ, f ) = EndF (V ), σq , fq where (V, q) is a 6-dimen-

sional quadratic space over a ¬eld F of characteristic di¬erent from 2, and suppose

q is isotropic. Suppose that disc q = ± · F —2 , so that the center of C0 (V, q) is

isomorphic to F [X]/(X 2 ’ ±). Then multiplying q by a suitable scalar, we may

assume that q has a diagonalization of the form

q = 1, ’1, ±, ’β, ’γ, βγ

for some β, γ ∈ F — . Let (e1 , . . . , e6 ) be an orthogonal basis of V which yields

the diagonalization above. In C0 (V, q), the elements e1 · e4 and e1 · e5 generate a

quaternion algebra (β, γ)F . The elements e1 · e4 · e5 · e6 and e1 · e4 · e5 · e2 centralize

this algebra and generate a split quaternion algebra (βγ)2 , ’βγ F ; therefore,

M2 (β, γ)F — F [X]/(X 2 ’ ±)

C0 (V, q)

by the double centralizer theorem (see (??)), and Proposition (??) shows that the

canonical involution „0 on C0 (V, q) is hyperbolic.

There is a corresponding result in characteristic 2: if the nonsingular 6-dimen-

sional quadratic space (V, q) is isotropic, we may assume (after scaling) that

q = [0, 0] ⊥ [1, ± + β] ⊥ γ [1, β] = [0, 0] ⊥ [1, ± + β] ⊥ [γ, βγ ’1 ]

for some ±, β ∈ F , γ ∈ F — . Thus, disc q = ± + „˜(F ), hence the center of C0 (V, q) is

isomorphic to F [X]/(X 2 + X + ±). Let (e1 , . . . , e6 ) be a basis of V which yields the

decomposition above. In C0 (V, q), the elements r = (e1 +e2 )·e3 and s = (e1 +e2 )·e4

satisfy r2 = 1, s2 = ± + β and rs + sr = 1, hence they generate a split quaternion

algebra 1, ± + β F (see §??). The elements (e1 +e2 )·e4 and (e1 +e2 )·e5 centralize

this algebra and generate a quaternion algebra γ, βγ ’1 F [β, γ)F . Therefore,

M2 [β, γ)F — F [X]/(X 2 + X + ±).

C0 (V, q)

As above, Proposition (??) shows that the canonical involution „0 is hyperbolic.

(15.36) Corollary. Let (B, „ ) ∈ A3 , with „ hyperbolic.

(1) Suppose the center Z(B) is a ¬eld, hence B is Brauer-equivalent to a quaternion

algebra (so ind B = 1 or 2); then the discriminant algebra D(B, „ ) splits and its

canonical quadratic pair („ , fD ) is associated with an isotropic quadratic form q.

The Witt index of q is 1 if ind B = 2; it is 2 if ind B = 1.

(2) Suppose Z(B) F — F , so that B E — E op for some central simple F -algebra

E of degree 4. If E is Brauer-equivalent to a quaternion algebra, then D(B, „ ) splits

and its canonical quadratic pair („ , fD ) is associated with an isotropic quadratic

form q. The Witt index of q is 1 if ind E = 2; it is 2 if ind E = 1.

Proof : (??) Since B has an involution of the second kind, Proposition (??) shows

that the Brauer-equivalent quaternion algebra has a descent to F . We may thus

assume that

√

for some ±, β, γ ∈ F — if char F = 2,

M2 (β, γ)F — F ( ±)

B

M2 [β, γ)F — F „˜’1 (±) for some ±, β ∈ F , γ ∈ F — if char F = 2,

230 IV. ALGEBRAS OF DEGREE FOUR

hence, by (??), (B, „ ) C0 (V, q), „0 where

1, ’1, ±, ’β, ’γ, βγ if char F = 2;

q

[0, 0] ⊥ [1, ± + β] ⊥ γ [1, β] if char F = 2.

If w(V, q) = 2, then Corollary (??) shows that 1 ∈ ind C0 (V, q), „0 , hence B is

split. Conversely, if B is split, then we may assume that γ = 1, and it follows that

w(V, q) = 2.

(??) The hypothesis yields

M2 (β, γ)F — F [X]/(X 2 ’ 1) for some β, γ ∈ F — if char F = 2,

B

M2 [β, γ)F — F [X]/(X 2 ’ X) for some β ∈ F , γ ∈ F — if char F = 2,

hence (B, „ ) C0 (V, q), „0 where

1, ’1, 1, ’β, ’γ, βγ if char F = 2;

q

[0, 0] ⊥ 1, γ [1, β] if char F = 2.

Since q is the orthogonal sum of a hyperbolic plane and the norm form of the

quaternion algebra Brauer-equivalent to E, we have w(V, q) = 1 if and only if

ind E = 2.

We next consider the case where the algebra A is not split. Since deg A = 6,

we must have ind A = 2, by (??). We write Z(A, σ, f ) for the center of the Cli¬ord

algebra C(A, σ, f ).

(15.37) Proposition. Let (A, σ, f ) ∈ D3 with ind A = 2.

(1) If the quadratic pair (σ, f ) is isotropic, then Z(A, σ, f ) is a splitting ¬eld of A.

(2) For each separable quadratic splitting ¬eld Z of A, there is, up to conjuga-

Z. If d ∈ F — is

tion, a unique quadratic pair (σ, f ) on A such that Z(A, σ, f )

such that the quaternion algebra Brauer-equivalent to A has the form (Z, d) F , then

C(A, σ, f ) M4 (Z) and the canonical involution σ is the adjoint involution with re-

spect to the 4-dimensional hermitian form on Z with diagonalization 1, ’1, 1, ’d .

Proof : (??) Let I ‚ A be a nonzero isotropic right ideal. We have rdim I ≥

1

2 deg A = 3, hence rdim I = 2 since ind A divides the reduced dimension of every

right ideal. Let e be an idempotent such that I = eA. As in the proof of (??),

we may assume that eσ(e) = σ(e)e = 0, hence e + σ(e) is an idempotent. Let

e1 = e + σ(e) and e2 = 1 ’ e1 ; then e1 A = eA • σ(e)A, hence rdim e1 A = 4

and therefore rdim e2 A = 2. Let Ai = ei Aei and let (σi , fi ) be the restriction of

the quadratic pair (σ, f ) to Ai for i = 1, 2. By (??), we have deg A1 = 4 and

deg A2 = 2, hence A2 is a quaternion algebra Brauer-equivalent to A. Moreover,

by (??) (if char F = 2) or (??) (if char F = 2),

disc(σ1 , f1 ) disc(σ2 , f2 ) if char F = 2,

disc(σ, f ) =

disc(σ1 , f1 ) + disc(σ2 , f2 ) if char F = 2.

Since eAe1 is an isotropic right ideal of reduced dimension 2 in A1 , the quadratic

pair (σ1 , f1 ) is hyperbolic, and Proposition (??) shows that its discriminant is

trivial. Therefore, disc(σ, f ) = disc(σ2 , f2 ), hence Z(A, σ, f ) Z(A2 , σ2 , f2 ). If

char F = 2, it was observed in (??) that Z(A2 , σ2 , f2 ) splits A2 , hence Z(A, σ, f )

splits A. To see that the same property holds if char F = 2, pick ∈ A2 such

that f2 (s) = TrdA2 ( s) for all s ∈ Sym(A2 , σ2 ); then TrdA2 ( ) = 1 and SrdA2 ( ) =

§15. EXCEPTIONAL ISOMORPHISMS 231

NrdA2 ( ) represents disc(σ2 , f2 ) in F/„˜(F ), so Z(A2 , σ2 , f2 ) F ( ). This completes

the proof of (??).

(??) Let Z be a separable quadratic splitting ¬eld of A and let d ∈ F — be such

that A is Brauer-equivalent to the quaternion algebra (Z, d)F , which we denote

simply by Q. We then have A M3 (Q). To prove the existence of an isotropic

quadratic pair (σ, f ) on A such that Z(A, σ, f ) Z, start with a quadratic pair

(θ, f1 ) on Q such that Z(Q, θ, f1 ) Z, and let (σ, f ) = (θ — ρ, f1— ) on A = Q —

M3 (F ), where ρ is the adjoint involution with respect to an isotropic 3-dimensional

bilinear form. We may choose for instance ρ = Int(u) —¦ t where

«

010

u = 1 0 0 ;

001

the involution σ is then explicitly de¬ned by

«

θ(x22 ) θ(x12 ) θ(x32 )

θ(x21 ) θ(x11 ) θ(x31 )

σ (xij )1¤i,j¤3 =

θ(x23 ) θ(x13 ) θ(x33 )

and the linear form f by

«

x11 s12 x13

s21 x23 = TrdQ (x11 ) + f1 (s33 ),

θ(x11 )

f

θ(x23 ) θ(x13 ) s33

for x11 , x13 , x23 ∈ Q and s12 , s21 , s33 ∈ Sym(Q, θ).

It is readily veri¬ed that

±«

x1 x2 x3

I = 0 0

0 x1 , x 2 , x 3 ∈ Q

0 0 0

is an isotropic right ideal, and that disc(A, σ, f ) = disc(Q, θ, f1 ), hence Z(A, σ, f )

Z.

Let (B, „ ) = C(A, σ, f ), σ . Since rdim I = 2, we have 2 ∈ ind(A, σ, f ),

hence Corollary (??) yields 1 ∈ ind(B, „ ). This relation shows that B is split,

hence B M4 (Z), and „ is the adjoint involution with respect to an isotropic

4-dimensional hermitian form h over Z. Multiplying h by a suitable scalar, we may

assume that h has a diagonalization 1, ’1, 1, ’a for some a ∈ F — . Corollary (??)

then shows that D(B, „ ) is Brauer-equivalent to the quaternion algebra (Z, a) F .

Since D(B, „ ) A, we have (Z, d)F (Z, a)F , hence a ≡ d mod N (Z/F ) and

therefore

h 1, ’1, 1, ’d .

The same arguments apply to every isotropic quadratic pair (σ, f ) on A such

that Z(A, σ, f ) Z: for every such quadratic pair, we have C(A, σ, f ), σ

M4 (Z), σh where h 1, ’1, 1, ’d , hence also

(A, σ, f ) D M4 (Z), σh , σh , fD .

This proves uniqueness of the quadratic pair (σ, f ) up to conjugation.

232 IV. ALGEBRAS OF DEGREE FOUR

Indices. Let (B, „ ) ∈ A3 and (A, σ, f ) ∈ D3 correspond to each other un-

der the equivalence A3 ≡ D3 . Let K be the center of B, which is isomorphic to

F ( disc(σ, f )) if char F = 2 and to F „˜’1 disc(σ, f ) if char F = 2. Our goal

is to relate the indices ind(A, σ, f ) and ind(B, „ ). For clarity, we consider the case

where K F — F separately.

(15.38) Proposition. Suppose K F — F , hence (B, „ ) (E — E op , µ) for some

central simple F -algebra E, where µ is the exchange involution. The only possibili-

ties for ind(A, σ, f ) are

{0}, {0, 1} and {0, 1, 2, 3}.

Moreover,

ind(A, σ, f ) = {0} ⇐’ ind(B, „ ) = {0} ⇐’ ind E = 4,

ind(A, σ, f ) = {0, 1} ⇐’ ind(B, „ ) = {0, 2} ⇐’ ind E = 2,

ind(A, σ, f ) = {0, 1, 2, 3} ⇐’ ind(B, „ ) = {0, 1, 2} ⇐’ ind E = 1.

Proof : Since deg A = 6, we have ind(A, σ, f ) ‚ {0, 1, 2, 3}. If 3 ∈ ind(A, σ, f ), then

A splits since ind A is a power of 2 which divides all the integers in ind(A, σ, f ).

In that case, we have (A, σ, f ) EndF (V ), σq , fq for some hyperbolic quadratic

space (V, q), hence ind(A, σ, f ) = {0, 1, 2, 3}.

Since K F — F , Proposition (??) shows that ind(A, σ, f ) = {0, 2}. There-

fore, if 2 ∈ ind(A, σ, f ), we must also have 1 or 3 ∈ ind(A, σ, f ), hence, as above,

(A, σ, f ) EndF (V ), σq , fq for some quadratic space (V, q) with w(V, q) ≥ 2.

Since disc(σ, f ) = disc q is trivial, the inequality w(V, q) ≥ 2 implies q is hyperbolic,

hence ind(A, σ, f ) = {0, 1, 2, 3}. Therefore, the only possibilities for ind(A, σ, f ) are

those listed above.

The relations between ind(B, „ ) and ind E readily follow from the de¬nition of

ind(E — E op , µ), and the equivalences ind(A, σ, f ) = {0, 1} ⇐’ ind E = 2 and

ind(A, σ, f ) = {0, 1, 2, 3} ⇐’ ind E = 1 follow from (??) and (??).

(15.39) Proposition. Suppose K is a ¬eld. The only possibilities for ind(A, σ, f )

are

{0}, {0, 1}, {0, 2} and {0, 1, 2}.

Moreover,

ind(A, σ, f ) = {0} ⇐’ ind(B, „ ) = {0},

ind(A, σ, f ) = {0, 1} ⇐’ ind(B, „ ) = {0, 2},

ind(A, σ, f ) = {0, 2} ⇐’ ind(B, „ ) = {0, 1},

ind(A, σ, f ) = {0, 1, 2} ⇐’ ind(B, „ ) = {0, 1, 2}.

Proof : If 3 ∈ ind(A, σ, f ), then (σ, f ) is hyperbolic, hence its discriminant is trivial,

by (??). This contradicts the hypothesis that K is a ¬eld. Therefore, we have

ind(A, σ, f ) ‚ {0, 1, 2}.

To prove the correspondence between ind(A, σ, f ) and ind(B, „ ), it now su¬ces

to show that 1 ∈ ind(A, σ, f ) if and only if 2 ∈ ind(B, „ ) and that 2 ∈ ind(A, σ, f )

if and only if 1 ∈ ind(B, „ ).

§16. BIQUATERNION ALGEBRAS 233

If 1 ∈ ind(A, σ, f ), then A is split, and Proposition (??) shows that 2 ∈

ind(B, „ ). Conversely, if 2 ∈ ind(B, „ ), then B is Brauer-equivalent to a quater-

nion algebra and „ is hyperbolic. It follows from (??) that 1 ∈ ind(A, σ, f ) in that

case. If 2 ∈ ind(A, σ, f ), then Corollary (??) yields 1 ∈ ind(B, „ ). Conversely, if

1 ∈ ind(B, „ ), then B splits and „ is the adjoint involution with respect to some

isotropic 4-dimensional hermitian form. By (??) (if „ is not hyperbolic) or (??) (if

„ is hyperbolic), it follows that 2 ∈ ind(A, σ, f ).

(15.40) Remark. The correspondence between ind(A, σ, f ) and ind(B, „ ) may be

summarized in the following relations (which hold when K F — F as well as when

K is a ¬eld):

1 ∈ ind(A, σ, f ) ⇐’ 2 ∈ ind(B, „ ), 2 ∈ ind(A, σ, f ) ⇐’ 1 ∈ ind(B, „ ),

3 ∈ ind(A, σ, f ) ⇐’ ind(A, σ, f ) = {0, 1, 2, 3}

⇐’ ind(B, „ ) = {0, 1, 2} and K F —F .

§16. Biquaternion Algebras

Algebras which are tensor products of two quaternion algebras are called bi-

quaternion algebras. Such algebras are central simple of degree 4 and exponent 2

(or 1). Albert proved the converse:

(16.1) Theorem (Albert [?, p. 369]). Every central simple algebra of degree 4 and

exponent 2 is a biquaternion algebra.

We present three proofs. The ¬rst two proofs rely heavily on the results of §??,

whereas the third proof, due to Racine, is more self-contained.

Throughout this section, A is a central simple algebra of degree 4 and exponent

1 or 2 over an arbitrary ¬eld F .

C ± (V, q) for some 6-dimensional

First proof (based on A3 ≡ D3 ): By (??), A

quadratic space (V, q) of trivial discriminant. The result follows from the structure

of Cli¬ord algebras of quadratic spaces.

Explicitly, if char F = 2 we may assume (after a suitable scaling) that q has a

diagonalization of the form

q = a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2

for some a1 , b1 , a2 , b2 ∈ F — . Let (e1 , . . . , e6 ) be an orthogonal basis of V which

yields that diagonalization. The even Cli¬ord algebra has a decomposition

C0 (V, q) = Q1 —F Q2 —F Z

where Q1 is the F -subalgebra generated by e1 ·e2 and e1 ·e3 , Q2 is the F -subalgebra

generated by e4 · e5 and e4 · e6 , and Z = F · 1 • F · e1 · e2 · e3 · e4 · e5 · e6 is the

F — F , hence C + (V, q) C ’ (V, q)

center of C0 (V, q). We have Z Q 1 — Q2 .

’1 ’1

Moreover, (1, b1 e2 · e3 , a1 e1 · e3 , e1 · e2 ) is a quaternion basis of Q1 which shows

Q1 (a1 , b1 )F , and (1, b’1 e5 · e6 , a’1 e4 · e6 , e5 · e6 ) is a quaternion basis of Q2 which

2 2

shows Q2 (a2 , b2 )F . Therefore, C ± (V, q) is a biquaternion algebra.

Similar arguments hold when char F = 2. We may then assume

q = [1, a1 b1 + a2 b2 ] ⊥ [a1 , b1 ] ⊥ [a2 , b2 ]

for some a1 , b1 , a2 , b2 ∈ F . Let (e1 , . . . , e6 ) be a basis of V which yields that

decomposition. In C0 (V, q), the elements e1 · e3 and e1 · e4 (resp. e1 · e5 and e1 · e6 )

234 IV. ALGEBRAS OF DEGREE FOUR

generate a quaternion F -algebra Q1 a 1 , b1 (resp. Q2 a 2 , b2 F ). We have

F

a decomposition

C0 (V, q) = Q1 —F Q2 —F Z

where Z = F · 1 • F · (e1 · e2 + e3 · e4 + e5 · e6 ) is the center of C0 (V, q). Since

Z F — F , it follows that

C + (V, q) C ’ (V, q) Q 1 —F Q2 a 1 , b1 — a 2 , b2 .

F F

Second proof (based on B2 ≡ C2 ): By (??) and (??), the algebra A carries an in-

volution σ of symplectic type. In the notation of §??, we have (A, σ) ∈ C2 . The

proof of the equivalence B2 ≡ C2 in (??) shows that A is isomorphic to the even

Cli¬ord algebra of some nonsingular 5-dimensional quadratic form:

C0 Symd(A, σ)0 , sσ .

(A, σ)

The result follows from the fact that even Cli¬ord algebras of odd-dimensional

quadratic spaces are tensor products of quaternion algebras (Scharlau [?, Theo-

rem 9.2.10]).

Third proof (Racine [?]): If A is not a division algebra, the theorem readily follows

from Wedderburn™s theorem (??), which yields a decomposition:

A M2 (F ) —F Q

for some quaternion algebra Q. We may thus assume that A is a division algebra.

Our ¬rst aim is to ¬nd in A a separable quadratic extension K of F . By (??),

A carries an involution σ. If char F = 2, we may start with any nonzero element

u ∈ Skew(A, σ); then u2 ∈ Sym(A, σ), hence F (u2 ) F (u). Since [F (u) : F ] = 4

or 2, we get [F (u ) F ] = 2 or 1 respectively. We choose K = F (u2 ) in the

2:

¬rst case and K = F (u) in the second case. In arbitrary characteristic, one may

choose a symplectic involution σ on A and take for K any proper extension of F

in Symd(A, σ) which is not contained in Symd(A, σ)0 , by (??).

The theorem then follows from the following proposition, which also holds when

A is not a division algebra. Recall that for every ´tale quadratic F -algebra K with

e

—

nontrivial automorphism ι and for every a ∈ F , the symbol (K, a)F stays for the

quaternion F -algebra K • Kz where multiplication is de¬ned by zx = ι(x)z for

x ∈ K and z 2 = a.

(16.2) Proposition. Suppose K is an ´tale quadratic F -algebra contained in a

e

central simple F -algebra A of degree 4 and exponent 2. There exist an a ∈ F — and

a quaternion F -algebra Q such that

A (K, a)F — Q.

Proof : If K is not a ¬eld, then A is not a division algebra, hence Wedderburn™s

theorem (??) yields

A (K, 1)F — Q

for some quaternion F -algebra Q.

If K is a ¬eld, the nontrivial automorphism ι extends to an involution „ on A

by (??). The restriction of „ to the centralizer B of K in A is an involution of

§16. BIQUATERNION ALGEBRAS 235

the second kind. Since B is a quaternion algebra over K, Proposition (??) yields a

quaternion F -algebra Q ‚ B such that

B = Q —F K.

By (??), there is a decomposition

B = Q — F CB Q

where CB Q is the centralizer of Q in B. This centralizer is a quaternion algebra

which contains K, hence

CB Q (K, a)F

for some a ∈ F — . We thus get the required decomposition.

16.A. Albert forms. Let A be a biquaternion algebra over a ¬eld F of ar-

bitrary characteristic. The algebra »2 A is split of degree 6 and carries a canonical

quadratic pair (γ, f ) of trivial discriminant (see (??)). Therefore, there are quad-

ratic spaces (V, q) of dimension 6 and trivial discriminant such that

(»2 A, γ, f ) EndF (V ), σq , fq

where (σq , fq ) is the quadratic pair associated with q.

(16.3) Proposition. For a biquaternion algebra A and a 6-dimensional quadratic

space (V, q) of discriminant 1, the following conditions are equivalent:

(1) (»2 A, γ, f ) EndF (V ), σq , fq ;

(2) A — A C0 (V, q);

(3) M2 (A) C(V, q).

Moreover, if (V, q) and (V , q ) are 6-dimensional quadratic spaces of discriminant 1

which satisfy these conditions for a given biquaternion algebra A, then (V, q) and

» · q for some » ∈ F — .

(V , q ) are similar, i.e., q

The quadratic forms which satisfy the conditions of this proposition are called

Albert forms of the biquaternion algebra A (and the quadratic space (V, q) is called

an Albert quadratic space of A). As the proposition shows, an Albert form is

determined only up to similarity by A. By contrast, it is clear from condition (??)

or (??) that any quadratic form of dimension 6 and discriminant 1 is an Albert

form for some biquaternion algebra A, uniquely determined up to isomorphism.

Proof : (??) ’ (??) Condition (??) implies that C(»2 A, γ, f ) C0 (V, q). Since

2 op op

(??) shows that C(» A, γ, f ) A — A and since A A , we get (??).

(??) ’ (??) Since the canonical involution σq on C0 (V, q) = C EndF (V ), σq , fq

is of the second kind, we derive from (??):

(A — Aop , µ),

C EndF (V ), σq , fq , σq

where µ is the exchange involution. By comparing the discriminant algebras of both

sides, we obtain

(»2 A, γ, f ).

D C EndF (V ), σq , fq , σq , σq , fD

Corollary (??) (or Theorem (??)) shows that there is a natural transformation

D —¦ C ∼ IdD3 , hence

=

D C EndF (V ), σq , fq , σq , σq , fD EndF (V ), σq , fq

236 IV. ALGEBRAS OF DEGREE FOUR

and we get (??).

(??) ” (??) This follows from the structure of Cli¬ord algebras of quadratic

forms: see for instance Lam [?, Ch. 5, Theorem 2.5] if char F = 2; similar arguments

hold in characteristic 2.

Finally, if (V, q) and (V , q ) both satisfy (??), then

EndF (V ), σq , fq EndF (V ), σq , fq ,

hence (V, q) and (V , q ) are similar, by (??).

(16.4) Example. Suppose char F = 2. For any a1 , b1 , a2 , b2 ∈ F — , the quadratic

form

q = a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2

is an Albert form of the biquaternion algebra (a1 , b1 )F — (a2 , b2 )F . This follows

from the computation of the Cli¬ord algebra C(q). (See the ¬rst proof of (??); see

also (??) below.)

Similarly, if char F = 2, then for any a1 , b1 , a2 , b2 ∈ F , the quadratic form

[1, a1 b1 + a2 b2 ] ⊥ [a1 , b1 ] ⊥ [a2 , b2 ]

is an Albert form of the biquaternion algebra a1 , b1 — a 2 , b2 F, and, for a1 ,

F

a2 ∈ F , b1 , b2 ∈ F — , the quadratic form

[1, a1 + a2 ] ⊥ b1 [1, a1 ] ⊥ b2 [1, a2 ]

is an Albert form of [a1 , b1 )F — [a2 , b2 )F .

Albert™s purpose in associating a quadratic form to a biquaternion algebra

was to obtain a necessary and su¬cient quadratic form theoretic criterion for the

biquaternion algebra to be a division algebra.

(16.5) Theorem (Albert [?]). Let A be a biquaternion algebra and let q be an

Albert form of A. The (Schur ) index of A, ind A, and the Witt index w(q) are

related as follows:

ind A = 4 w(q) = 0;

if and only if

(in other words, A is a division algebra if and only if q is anisotropic);

ind A = 2 w(q) = 1;

if and only if

ind A = 1 w(q) = 3;

if and only if

(in other words, A is split if and only if q is hyperbolic).

Proof : This is a particular case of (??).

Another relation between biquaternion algebras and their Albert forms is the

following:

(16.6) Proposition. The multipliers of similitudes of an Albert form q of a bi-

quaternion algebra A are given by

G(q) = F —2 · NrdA (A— )

and the spinor norms by

Sn(q) = { » ∈ F — | »2 ∈ NrdA (A— ) }.

Proof : This is a direct application of (??).

§16. BIQUATERNION ALGEBRAS 237

Even though there is no canonical choice for an Albert quadratic space of a

biquaternion algebra A, when an involution of the ¬rst kind σ on A is ¬xed, an

Albert form may be de¬ned on the vector space Symd(A, σ) of symmetric elements if

σ is symplectic and on the vector space Skew(A, σ) if σ is orthogonal and char F = 2.

Moreover, Albert forms may be used to de¬ne an invariant of symplectic involutions,

as we now show.

16.B. Albert forms and symplectic involutions. Let σ be a symplectic

involution on the biquaternion F -algebra A. Recall from §?? (see (??)) that the

reduced characteristic polynomial of every symmetrized element is a square:

2

PrdA,s (X) = Prpσ,s (X)2 = X 2 ’ Trpσ (s)X + Nrpσ (s) for s ∈ Symd(A, σ).

Since deg A = 4, the polynomial map Nrpσ : Symd(A, σ) ’ F has degree 2. We

show in (??) below that Symd(A, σ), Nrpσ is an Albert quadratic space of A.

A key tool in this proof is the linear endomorphism of Symd(A, σ) de¬ned by

x = Trpσ (x) ’ x for x ∈ Symd(A, σ).

Since Prpσ,x (x) = 0 for all x ∈ Symd(A, σ), we have

Nrpσ (x) = xx = xx for x ∈ Symd(A, σ).

(16.7) Lemma. For x ∈ Symd(A, σ) and a ∈ A— ,

axσ(a) = NrdA (a)σ(a)’1 xa’1 .

Proof : Since both sides of the equality above are linear in x, it su¬ces to show that

the equality holds for x in some basis of Symd(A, σ). It is readily seen by scalar

extension to a splitting ¬eld that Symd(A, σ) is spanned by invertible elements.

Therefore, it su¬ces to prove the equality for invertible x. In that case, the property

follows by comparing the equalities

NrdA (a) Nrpσ (x) = Nrpσ axσ(a) = axσ(a) · axσ(a)

and

Nrpσ (x) = axσ(a) · σ(a)’1 xa’1 .

(16.8) Proposition. The quadratic space Symd(A, σ), Nrpσ is an Albert quad-

ratic space of A.

Proof : For x ∈ Symd(A, σ), let i(x) = x x ∈ M2 (A). Since xx = xx = Nrpσ (x),

0

0

we have i(x)2 = Nrpσ (x), hence the universal property of Cli¬ord algebras shows

that i induces an F -algebra homomorphism

(16.9) i— : C Sym(A, σ), Nrpσ ’ M2 (A).

This homomorphism is injective since C Sym(A, σ), Nrpσ is simple, and it is sur-

jective by dimension count.

(16.10) Remark. Proposition (??) shows that A contains a right ideal of reduced

dimension 2 if and only if the quadratic form Nrpσ is isotropic. Therefore, A is a

division algebra if and only if Nrpσ is anisotropic; in view of (??), this observation

yields an alternate proof of a (substantial) part of Albert™s theorem (??).

238 IV. ALGEBRAS OF DEGREE FOUR

The isomorphism i— of (??) may also be used to give an explicit description of

the similitudes of the Albert quadratic space Symd(A, σ), Nrpσ , thus yielding an

alternative proof of the relation between Cli¬ord groups and symplectic similitudes

in (??).

(16.11) Proposition. The proper similitudes of Symd(A, σ), Nrpσ are of the

form

x ’ »’1 axσ(a)

where » ∈ F — and a ∈ A— .

The improper similitudes of Symd(A, σ), Nrpσ are of the form

x ’ »’1 axσ(a)

where » ∈ F — and a ∈ A— . The multiplier of these similitudes is »’2 NrdA (a).

Proof : Since Nrpσ (x) = Nrpσ (x) for all x ∈ Sym(A, σ), it follows from (??) that

the maps above are similitudes with multiplier »’2 NrdA (a) for all » ∈ F — and

a ∈ A— .

Conversely, let f ∈ GO Symd(A, σ), Nrpσ be a similitude and let ± = µ(f )

be its multiplier. The universal property of Cli¬ord algebras shows that there is an

isomorphism

f— : C Symd(A, σ), Nrpσ ’ M2 (A)

0 ±’1 f (x)

which maps x ∈ Symd(A, σ) to . By comparing f— with the isomor-

f (x) 0

phism i— of (??), we get an automorphism f— —¦i’1 of M2 (A). Note that the checker-

—

board grading of M2 (A) corresponds to the canonical Cli¬ord algebra grading under

both f— and i— . Therefore, f— —¦ i’1 is a graded automorphism, and f— —¦ i’1 = Int(u)

— —

for some u ∈ GL2 (A) of the form

v0 0 v

u= or .

0w w 0

Moreover, inspection shows that the automorphism C(f ) of C0 Symd(A, σ), Nrpσ

induced by f ¬ts in the commutative diagram

C(f )

C0 (Nrpσ ) ’ ’ ’ C0 (Nrpσ )

’’

¦ ¦

¦ ¦i

i— —

f— —¦i’1

’ ’—’

A—A ’’ A—A

where we view A—A as A A ‚ M2 (A). Therefore, in view of (??), the similitude f

0

0

is proper if and only if f— —¦ i’1 maps each component of A — A into itself. This

—

means that f is proper if u = v w and improper if u = w v .

0 0

0 0

’1

In particular, if f (x) = » axσ(a) for x ∈ Symd(A, σ), then

» NrdA (a)’1 axσ(a)

0

f— —¦ i’1 i(x) =

— ’1

» axσ(a) 0

and (??) shows that the right side is

»σ(a)’1 »’1 σ(a) »σ(a)’1

0 0x 0 0

· · = Int i(x) .

’1

0 a x0 0 a 0 a

§16. BIQUATERNION ALGEBRAS 239

Since the matrices of the form i(x), x ∈ Symd(A, σ) generate M2 (A) (as Symd(A, σ)

generates C Symd(A, σ), Nrpσ ), it follows that

»σ(a)’1 0

i’1

f— —¦ = Int ,

— 0 a

hence f is proper. Similarly, the same arguments show that the similitudes x ’

»’1 axσ(a) are improper.

Returning to the case where f is an arbitrary similitude of Symd(A, σ), Nrpσ

and f— —¦ i’1 = Int(u) with u = v w or w v , we apply f— —¦ i’1 to i(x) for

0 0

— —

0 0

x ∈ Symd(A, σ) and get

±’1 f (x) 0 x ’1

0

(16.12) =u u.

x0

f (x) 0

Comparing the lower left corners yields that

wxv ’1 if f is proper,

(16.13) f (x) =

wxv ’1 if f is improper.

Let θ be the involution on M2 (A) de¬ned by

a11 a12 σ(a22 ) ’σ(a12 )

θ = .

a21 a22 ’σ(a21 ) σ(a11 )

Applying θ to both sides of (??), we get

±’1 f (x) 0x

0

= θ(u)’1 θ(u).

x0

f (x) 0

Therefore, θ(u)u commutes with the matrices of the form i(x) for x ∈ Symd(A, σ).

Since these matrices generate M2 (A), it follows that θ(u)u ∈ F — , hence

σ(w)v = σ(v)w ∈ F — .

Letting σ(w)v = », we derive from (??) that

»’1 wxσ(w) if f is proper,

f (x) =

»’1 wxσ(w) if f is improper.

Since the multipliers of the similitudes x ’ »’1 axσ(a) and x ’ »’1 axσ(a)

are »’2 NrdA (a), the multipliers of the Albert form Nrpσ are

G(Nrpσ ) = F —2 · NrdA (A— ).

We thus get another proof of the ¬rst part of (??).

(16.14) Example. Suppose Q is a quaternion algebra over F , with canonical in-

volution γ, and A = M2 (Q) with the involution σ de¬ned by σ (qij )1¤i,j¤2 =

t

γ(qij ) 1¤i,j¤2 ; then

±11 a12

Symd(A, σ) = ±11 , ±22 ∈ F , a12 ∈ Q .

γ(a12 ) ±22

±11 a12

For a = ∈ Symd(A, σ), we have Trpσ (a) = ±11 + ±22 , hence a =

γ(a12 ) ±22

±22 ’a12

and therefore

’γ(a12 ) ±11

Nrpσ (a) = ±11 ±22 ’ NrdQ (a12 ).

240 IV. ALGEBRAS OF DEGREE FOUR

This expression is the Moore determinant of the hermitian matrix a (see Jacobson

[?]). This formula shows that the matrices in Symd(A, σ) whose diagonal entries

vanish form a quadratic space isometric to (Q, ’ NrdQ ). On the other hand, the

diagonal matrices form a hyperbolic plane H, and

H ⊥ ’ NrdQ .

Nrpσ

Also, for the involution θ de¬ned by

a11 a12 γ(a11 ) ’γ(a21 )

θ = ,

a21 a22 ’γ(a12 ) γ(a22 )

we have

±11 a12

Symd(A, θ) = ±11 , ±22 ∈ F , a12 ∈ Q .

’γ(a12 ) ±22

±11 a12

For a = ∈ Symd(A, σ), we get

’γ(a12 ) ±22

Nrpθ (a) = ±11 ±22 + NrdQ (a12 ),

hence

H ⊥ NrdQ .

Nrpθ

A more general example is given next.

(16.15) Example. Suppose A = Q1 — Q2 is a tensor product of quaternion al-

gebras Q1 , Q2 with canonical (symplectic) involutions γ1 , γ2 . Let v1 be a unit

in Skew(Q1 , γ1 ) and σ1 = Int(v1 ) —¦ γ1 . The involution σ1 on Q1 is orthogonal,

unless v1 ∈ F — , a case which occurs only if char F = 2. Therefore, the involution

σ = σ1 — γ2 on A is symplectic in all cases, by (??). Our goal is to compute

explicitly the quadratic form Nrpσ .

As a ¬rst step, observe that

Alt(A, γ1 — γ2 ) = { x1 — 1 ’ 1 — x2 | TrdQ1 (x1 ) = TrdQ2 (x2 ) },

as pointed out in Exercise ?? of Chapter ??; therefore,

Symd(A, σ) = (v1 — 1) · Alt(A, γ1 — γ2 )

= { v1 x1 — 1 ’ v1 — x2 | TrdQ1 (x1 ) = TrdQ2 (x2 ) }.

For x1 ∈ Q1 and x2 ∈ Q2 such that TrdQ1 (x1 ) = TrdQ2 (x2 ), there exist y1 ∈ Q1 ,

y2 ∈ Q2 such that x1 = TrdQ2 (y2 )y1 and x2 = TrdQ1 (y1 )y2 (see (??)), hence

x1 — 1 ’ 1 — x2 = y1 — γ2 (y2 ) ’ γ1 (y1 ) — y2

and therefore

v1 x1 — 1 ’ v1 — x2 = v1 y1 — γ2 (y2 ) + σ v1 y1 — γ2 (y2 ) .

By (??), it follows that

Trpσ (v1 x1 — 1 ’ v1 — x2 ) = TrdA v1 y1 — γ2 (y2 ) = TrdQ1 (v1 y1 ) TrdQ2 (y2 ).

Since TrdQ2 (y2 )y1 = x1 , we get

Trpσ (v1 x1 — 1 ’ v1 — x2 ) = TrdQ1 (v1 x1 ),

hence

v1 x1 — 1 ’ v1 — x2 = γ1 (v1 x1 ) — 1 + v1 — x2

§16. BIQUATERNION ALGEBRAS 241

and ¬nally

Nrpσ (v1 x1 — 1 ’ v1 — x2 ) = NrdQ1 (v1 ) NrdQ1 (x1 ) ’ NrdQ2 (x2 ) .

This shows that the form Nrpσ on Symd(A, σ) is similar to the quadratic form

qγ1 —γ2 on Alt(A, γ1 — γ2 ) de¬ned by

qγ1 —γ2 (x1 — 1 ’ 1 — x2 ) = NrdQ1 (x1 ) ’ NrdQ2 (x2 )

for x1 ∈ Q1 , x2 ∈ Q2 such that TrdQ1 (x1 ) = TrdQ2 (x2 ).

To give a more explicit description of qγ1 —γ2 , we consider the case where

char F = 2 separately. Suppose ¬rst char F = 2, and let Q1 = (a1 , b1 )F , Q2 =

(a2 , b2 )F with quaternion bases (1, i1 , j1 , k1 ) and (1, i2 , j2 , k2 ) respectively. Then

(i1 — 1, j1 — 1, k1 — 1, 1 — i2 , 1 — j2 , 1 — k2 ) is an orthogonal basis of Alt(A, γ1 — γ2 )

which yields the following diagonalization of qγ1 —γ2 :

qγ1 —γ2 = ’a1 , ’b1 , a1 b1 , a2 , b2 , ’a2 b2

(compare with (??) and (??)); therefore,

Nrpσ NrdQ1 (v1 ) · ’a1 , ’b1 , a1 b1 , a2 , b2 , ’a2 b2 .

Suppose next that char F = 2, and let Q1 = [a1 , b1 )F , Q2 = [a2 , b2 )F with quater-

nion bases (in characteristic 2) (1, u1 , v1 , w1 ) and (1, u2 , v2 , w2 ) respectively. A basis

of Alt(A, γ1 — γ2 ) is (1, u1 — 1 + 1 — u2 , v1 — 1, w1 — 1, 1 — v2 , 1 — w2 ). With respect

to this basis, the form qγ1 —γ2 has the following expression:

qγ1 —γ2 = [1, a1 + a2 ] ⊥ b1 · [1, a1 ] ⊥ b2 · [1, a2 ]

(compare with (??)); therefore,

Nrpσ NrdQ1 (v1 ) · [1, a1 + a2 ] ⊥ b1 · [1, a1 ] ⊥ b2 · [1, a2 ] .

The following proposition yields a decomposition of the type considered in the

example above for any biquaternion algebra with symplectic involution; it is thus

an explicit version of the second proof of (??). However, for simplicity we restrict to

symplectic involutions which are not hyperbolic.25 In view of (??), this hypothesis

means that the space

Symd(A, σ)0 = { x ∈ Symd(A, σ) | Trpσ (x) = 0 }

does not contain any nonzero vector x such that x2 = 0 or, equivalently, Nrpσ (x) =

0. Therefore, all the nonzero elements in Symd(A, σ)0 are invertible.

(16.16) Proposition. Let (A, σ) be a biquaternion algebra with symplectic invo-

lution over an arbitrary ¬eld F . Assume that σ is not hyperbolic, and let V ‚

Symd(A, σ) be a 3-dimensional subspace such that

F ‚ V ‚ Symd(A, σ)0 .

Then there exists a unique quaternion subalgebra Q1 ‚ A containing V . This

quaternion algebra is stable under σ, and the restriction σ1 = σ|Q1 is orthogonal.

Therefore, for Q2 = CA Q1 the centralizer of Q1 , we have

(A, σ) = (Q1 , σ1 ) — (Q2 , γ2 )

where γ2 is the canonical involution on Q2 .

25 See Exercise ?? of Chapter ?? for the hyperbolic case.

242 IV. ALGEBRAS OF DEGREE FOUR

Proof : Choose v ∈ V F such that Trpσ (v) = 0; then F [v] = F • vF is an ´talee

quadratic F -subalgebra of A, and restricts to the nontrivial automorphism of

F [v]. Pick a nonzero vector u ∈ V © Symd(A, σ)0 which is orthogonal to v for

the polar form bNrpσ ; we then have uv + vu = 0, which means that uv = vu,

since Trpσ (u) = 0. The hypothesis that σ is not hyperbolic ensures that u is

invertible; therefore, u and v generate a quaternion subalgebra Q1 = F [v], u2 F .

This quaternion subalgebra contains V , and is indeed generated by V . Since u and

v are symmetric under σ, it is stable under σ, and Sym(Q1 , σ1 ) = V . Moreover,

Sym(Q1 , σ1 ) contains the element v such that TrdQ1 (v) = v + v = Trpσ (v) = 0,

hence σ1 is orthogonal. The rest follows from (??) and (??).

The invariant of symplectic involutions. Let σ be a ¬xed symplectic in-

volution on a biquaternion algebra A. To every other symplectic involution „ on A,

we associate a quadratic form jσ („ ) over F which classi¬es symplectic involutions

up to conjugation: jσ („ ) jσ („ ) if and only if „ = Int(a) —¦ „ —¦ Int(a)’1 for some

a ∈ A— .

We ¬rst compare the Albert forms Nrpσ and Nrp„ associated with symplectic

involutions σ and „ . Recall from (??) that „ = Int(u) —¦ σ for some unit u ∈

Symd(A, σ). Multiplication on the left by the element u then de¬nes a linear map

∼

Symd(A, σ) ’ Symd(A, „ ).

’

(16.17) Lemma. For all x ∈ Symd(A, σ),

Nrp„ (ux) = Nrpσ (u) Nrpσ (x).

Proof : Both sides of the equation to be established are quadratic forms on the space

Symd(A, σ). These quadratic forms di¬er at most by a factor ’1, since squaring

both sides yields the equality

NrdA (ux) = NrdA (u) NrdA (x).

On the other hand, for x = 1 these quadratic forms take the same nonzero value

since from the fact that PrdA,u = Prp2 = Prp2 it follows that Prpσ,u = Prp„,u ,

σ,u „,u

hence Nrpσ (u) = Nrp„ (u). Therefore, the quadratic forms are equal.

Let W F denote the Witt ring of nonsingular bilinear forms over F and write

Wq F for the W F -module of even-dimensional nonsingular quadratic forms. For

every integer k, the k-th power of the fundamental ideal IF of even-dimensional

forms in W F is denoted I k F ; we write I k Wq F for the product I k F · Wq F . Thus, if

I k Wq F = I k+1 F if char F = 2. From the explicit formulas in (??), it is clear that

Albert forms are in IWq F ; indeed, if char F = 2,

a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2 = ’ a1 , b1 + a2 , b2 in W F,

and, if char F = 2,

[1, a1 + a2 ] ⊥ b1 · [1, a1 ] ⊥ b2 · [1, a2 ] = b1 , a1 ]] + b2 , a2 ]] in Wq F.

(16.18) Proposition. Let σ, „ be symplectic involutions on a biquaternion F -

algebra A and let „ = Int(u) —¦ σ for some unit u ∈ Symd(A, σ). In the Witt

group Wq F ,

Nrpσ (u) · Nrpσ = Nrpσ ’ Nrp„ .

There is a 3-fold P¬ster form jσ („ ) ∈ I 2 Wq F and a scalar » ∈ F — such that

» · jσ („ ) = Nrpσ (u) · Nrpσ in Wq F .

§16. BIQUATERNION ALGEBRAS 243

The P¬ster form jσ („ ) is uniquely determined by the condition

mod I 3 Wq F.

jσ („ ) ≡ Nrpσ (u) · Nrpσ

Proof : Lemma (??) shows that multiplication on the left by u is a similitude:

Symd(A, σ), Nrpσ ’ Symd(A, „ ), Nrp„

with multiplier Nrpσ (u). Therefore, Nrp„ Nrpσ (u) · Nrpσ , hence

Nrpσ ’ Nrp„ Nrpσ (u) · Nrpσ .

We next show the existence of the 3-fold P¬ster form jσ („ ). Since 1 and

u are anisotropic for Nrpσ , there exist nonsingular 3-dimensional subspaces U ‚

Symd(A, σ) containing 1 and u. Choose such a subspace and let qU be the restriction

of Nrpσ to U . Let q0 be a 4-dimensional form in IWq F containing qU as a subspace:

if char F = 2 and qU a1 , a2 , a3 , we set q0 = a1 , a2 , a3 , a1 a2 a3 ; if char F = 2

[a1 , a2 ] ⊥ [a3 ], we set q0 = [a1 , a2 ] ⊥ [a3 , a1 a2 a’1 ]. Since the quadratic

and qU 3

forms Nrpσ and q0 have isometric 3-dimensional subspaces, there is a 4-dimensional

quadratic form q1 such that

q0 + q1 = Nrpσ in Wq F.

The form q1 lies in IWq F , since q0 and Nrpσ are in this subgroup. Moreover, since

Nrpσ (u) is represented by qU , hence also by q0 , we have Nrpσ (u) ·q0 = 0 in Wq F .

Therefore, multiplying both sides of the equality above by Nrp σ (u) , we get

Nrpσ (u) · q1 = Nrpσ (u) · Nrpσ in Wq F.

The form on the left is a scalar multiple of a 3-fold P¬ster form which may be

chosen for jσ („ ). This P¬ster form satis¬es jσ („ ) ≡ Nrpσ (u) · q1 mod I 3 Wq F ,

hence also

mod I 3 Wq F.

jσ („ ) ≡ Nrpσ (u) · Nrpσ

It remains only to show that it is uniquely determined by this condition. This

follows from the following general observation: if π, π are k-fold P¬ster forms such

that

mod I k Wq F,

π≡π

then the di¬erence π ’ π is represented by a quadratic form of dimension 2k+1 ’ 2

since π and π both represent 1. On the other hand, π ’ π ∈ I k Wq F , hence the

Hauptsatz of Arason and P¬ster (see26 Lam [?, p. 289] or Scharlau [?, Ch. 4, §5])

shows that π ’ π = 0. Therefore, π π .

We next show that the invariant jσ classi¬es symplectic involutions up to con-

jugation:

(16.19) Theorem. Let σ, „ , „ be symplectic involutions on a biquaternion algebra

A over an arbitrary ¬eld F , and let „ = Int(u) —¦ σ, „ = Int(u ) —¦ σ for some units

u, u ∈ Symd(A, σ). The following conditions are equivalent:

(1) „ and „ are conjugate, i.e., there exists a ∈ A— such that „ = Int(a) —¦ „ —¦

Int(a)’1 ;

(2) Nrpσ (u) Nrpσ (u ) ∈ F —2 · Nrd(A— );

26 The proofs given there are easily adapted to the characteristic 2 case. The main ingredient

is the Cassels-P¬ster subform theorem, of which a characteristic 2 analogue is given in P¬ster [?,

Theorem 4.9, Chap. 1].

244 IV. ALGEBRAS OF DEGREE FOUR

(3) Nrpσ (u) Nrpσ (u ) ∈ G(Nrpσ );

(4) jσ („ ) jσ („ );

(5) Nrp„ Nrp„ .

Proof : (??) ’ (??) If there exists some a ∈ A— such that „ = Int(a)—¦„ —¦Int(a)’1 ,

then, since the right-hand side is also equal to Int a„ (a) —¦ „ , we get

Int(u ) = Int a„ (a)u = Int auσ(a) ,

hence u = »’1 auσ(a) for some » ∈ F — . By (??), it follows that

Nrpσ (u ) = »’2 NrdA (a) Nrpσ (u),

proving (??).

(??) ⇐’ (??) This readily follows from (??), since Nrpσ is an Albert form

of A.

(??) ’ (??) Suppose Nrpσ (u ) = µ Nrpσ (u) for some µ ∈ G(Nrpσ ). We may

then ¬nd a proper similitude g ∈ GO+ (Nrpσ ) with multiplier µ. Then

Nrpσ g(u) = µ Nrpσ (u) = Nrpσ (u ),

hence there is a proper isometry h ∈ O+ (Nrpσ ) such that h —¦ g(u) = u . By (??),

we may ¬nd an a ∈ A— and a » ∈ F — such that

h —¦ g(x) = »’1 axσ(a) for all x ∈ Symd(A, σ).

In particular, u = »’1 auσ(a), hence Int(a) —¦ „ = „ —¦ Int(a).

(??) ⇐’ (??) Since jσ („ ) and jσ („ ) are the unique 3-fold P¬ster forms which

are equivalent modulo I 3 Wq F to Nrpσ (u) · Nrpσ and Nrpσ (u ) · Nrpσ respec-

tively, we have jσ („ ) jσ („ ) if and only if Nrpσ (u) · Nrpσ ≡ Nrpσ (u ) · Nrpσ

mod I 3 Wq F . Using the relation Nrpσ (u) ’ Nrpσ (u ) ≡ Nrpσ (u) Nrpσ (u )

mod I 2 F , we may rephrase the latter condition as

Nrpσ (u) Nrpσ (u ) · Nrpσ ∈ I 3 Wq F.

By the Arason-P¬ster Hauptsatz, this relation holds if and only if

Nrpσ (u) Nrpσ (u ) · Nrpσ = 0,

which means that Nrpσ (u) Nrpσ (u ) ∈ G(Nrpσ ).

(??) ⇐’ (??) The relations

mod I 3 Wq F mod I 3 Wq F

jσ („ ) ≡ Nrpσ ’ Nrp„ and jσ („ ) ≡ Nrpσ ’ Nrp„

jσ („ ) if and only if Nrp„ ’ Nrp„ ∈ I 3 Wq F . By the Arason-

show that jσ („ )

P¬ster Hauptsatz, this relation holds if and only if Nrp„ ’ Nrp„ = 0.

(16.20) Remark. Theorem (??) shows that the conditions in (??) are also equiv-

alent to: Symd(A, „ )0 , s„ Symd(A, „ )0 , s„ .

(16.21) Example. As in (??), consider a quaternion F -algebra Q with canonical

involution γ, and A = M2 (Q) with the involution σ de¬ned by σ (qij )1¤i,j¤2 =

t

γ(qij ) . Let „ = Int(u) —¦ σ for some invertible matrix u ∈ Symd(A, σ). As

observed in (??), we have Nrpσ H ⊥ ’ NrdQ ; therefore,

jσ („ ) = Nrpσ (u) · NrdQ .

§16. BIQUATERNION ALGEBRAS 245

Since jσ („ ) is an invariant of „ , the image of Nrpσ (u) in F — / NrdQ (Q— ) also is an

invariant of „ up to conjugation. In fact, since Nrpσ (u) is the Moore determinant

of u, this image is the Jacobson determinant of the hermitian form

y1

h (x1 , x2 ), (y1 , y2 ) = γ(x1 ) γ(x2 ) · u ·

y2

on the 2-dimensional Q-vector space Q2 . (See the notes of Chapter ??.)

Of course, if Q is split, then NrdQ is hyperbolic, hence jσ („ ) = 0 for all sym-

plectic involutions σ, „ . Therefore, all the symplectic involutions are conjugate in

this case. (This is clear a priori, since all the symplectic involutions on a split

algebra are hyperbolic.)

16.C. Albert forms and orthogonal involutions. Let σ be an orthogonal

involution on a biquaternion F -algebra A. Mimicking the construction of the Albert

form associated to a symplectic involution, in this subsection we de¬ne a quadratic

form qσ on the space Skew(A, σ) in such a way that Skew(A, σ), qσ is an Albert

quadratic space. By contrast with the symplectic case, the form qσ is only de¬ned

up to a scalar factor, however, and our discussion is restricted to the case where the

characteristic is di¬erent from 2. We also show how the form qσ is related to the

norm form of the Cli¬ord algebra C(A, σ) and to the generalized pfa¬an de¬ned

in §??.

Throughout this subsection, we assume that char F = 2.

(16.22) Proposition. There exists a linear endomorphism

pσ : Skew(A, σ) ’ Skew(A, σ)

which satis¬es the following two conditions:

(1) xpσ (x) = pσ (x)x ∈ F for all x ∈ Skew(A, σ);

(2) an element x ∈ Skew(A, σ) is invertible if and only if xpσ (x) = 0.

The endomorphism pσ is uniquely determined up to a factor in F — . More precisely,

if pσ : Skew(A, σ) ’ Skew(A, σ) is a linear map such that xpσ (x) ∈ F for all

x ∈ Skew(A, σ) (or pσ (x)x ∈ F for all x ∈ Skew(A, σ)), then

pσ = »pσ

for some » ∈ F .

Proof : By (??), the intersection Skew(A, σ) © A— is nonempty. Let u be a skew-

symmetric unit and „ = Int(u) —¦ σ. The involution „ is symplectic by (??), and we

have

Sym(A, „ ) = u · Skew(A, σ) = Skew(A, σ) · u’1 .

Therefore, for x ∈ Skew(A, σ) we may consider ux ∈ Sym(A, „ ) where

: Sym(A, „ ) ’ Sym(A, „ )

is as in (??), and set

pσ (x) = uxu ∈ Skew(A, σ) for x ∈ Skew(A, σ).

We have pσ (x)x = uxux = Nrp„ (ux) ∈ F and

xpσ (x) = u’1 (uxux)u = u’1 Nrp„ (ux)u = Nrp„ (ux),

hence pσ satis¬es (??). It also satis¬es (??), since Nrp„ (ux)2 = NrdA (ux).

246 IV. ALGEBRAS OF DEGREE FOUR

In order to make this subsection independent of §??, we give an alternate proof

of the existence of pσ . Consider an arbitrary decomposition of A into a tensor

product of quaternion subalgebras:

A = Q 1 —F Q2

and let θ = γ1 — γ2 be the tensor product of the canonical (conjugation) involutions

on Q1 and Q2 . The involution θ is orthogonal since char F = 2, and

Skew(A, θ) = (Q0 — 1) • (1 — Q0 ),

1 2

where Q0 and Q0 are the spaces of pure quaternions in Q1 and Q2 respectively.

1 2

De¬ne a map pθ : Skew(A, θ) ’ Skew(A, θ) by

pθ (x1 — 1 + 1 — x2 ) = x1 — 1 ’ 1 — x2

for x1 ∈ Q0 and x2 ∈ Q0 . For x = x1 — 1 + 1 — x2 ∈ Skew(A, θ) we have

1 2

xpθ (x) = pθ (x)x = x2 ’ x2 = ’ NrdQ1 (x1 ) + NrdQ2 (x2 ) ∈ F,

1 2

hence (??) holds for pθ . If xpθ (x) = 0, then x is clearly invertible. Conversely,

if x is invertible and xpθ (x) = 0, then pθ (x) = 0, hence x = 0, a contradiction.

Therefore, pθ also satis¬es (??).

If σ is an arbitrary orthogonal involution on A, we have σ = Int(v) —¦ θ for some

v ∈ Sym(A, θ) © A— , by (??). We may then set

pσ (x) = vpθ (xv) for x ∈ Skew(A, σ)

and verify as above that pσ satis¬es the required conditions.

We next prove uniqueness of pσ up to a scalar factor. The following arguments

are based on Wadsworth [?]. For simplicity, we assume that F has more than three

elements; the result is easily checked when F = F3 .

Let pσ be a map satisfying (??) and (??), and let pσ : Skew(A, σ) ’ Skew(A, σ)

be such that xpσ (x) ∈ F for all x ∈ Skew(A, σ). For x ∈ Skew(A, σ), we let

qσ (x) = xpσ (x) ∈ F and qσ (x) = xpσ (x) ∈ F.

Let x ∈ Skew(A, σ) © A— ; we have pσ (x) = qσ (x)x’1 and pσ (x) = qσ (x)x’1 , hence

qσ (x)

pσ (x) = pσ (x).

qσ (x)

Suppose y is another unit in Skew(A, σ), and that it is not a scalar multiple of x.

We also have pσ (y) = qσ (y)y ’1 , hence pσ (y) is not a scalar multiple of pσ (x), and

q (y)

pσ (y) = qσ (y) pσ (y). We may ¬nd some ± ∈ F — such that x + ±y ∈ A— , since the

σ