for Q, the proof is complete.

15.C. B2 ≡ C2 . The arguments to prove the equivalence of B2 and C2 are

similar to those used in §?? to prove B1 ≡ C1 .

For any oriented quadratic space (V, q, ζ) ∈ B2 (of trivial discriminant), the even

Cli¬ord algebra C0 (V, q) is central simple of degree 4, and its canonical involution

„ (= σq ) is symplectic. We may therefore de¬ne a functor

C : B2 ’ C 2

by

C(V, q, ζ) = C0 (V, q), „ .

On the other hand, let (A, σ) be a central simple F -algebra of degree 4 with sym-

plectic involution. As observed in (??), the reduced characteristic polynomial of

every symmetrized element is a square; the pfa¬an trace Trpσ is a linear form on

V and the pfa¬an norm Nrpσ is a quadratic form on V such that

2

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

In particular, if a ∈ Symd(A, σ) is such that Trpσ (a) = 0, then a2 = ’ Nrpσ (a) ∈ F .

Let

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

= { a ∈ Sym(A, σ) | TrdA (a) = 0 } if char F = 2 .

This is a vector space of dimension 5 over F . The map sσ : Symd(A, σ)0 ’ F

de¬ned by

sσ (a) = a2

§15. EXCEPTIONAL ISOMORPHISMS 217

is a quadratic form on Symd(A, σ)0 . Inspection of the split case shows that this

form is nonsingular. By the universal property of Cli¬ord algebras, the inclusion

Symd(A, σ) ’ A induces an F -algebra homomorphism

hA : C Symd(A, σ)0 , sσ ’ A,

(15.15)

which is not injective since dimF C Symd(A, σ)0 , sσ = 25 while dimF A = 24 .

Therefore, the Cli¬ord algebra C Symd(A, σ)0 , sσ is not simple, hence the dis-

criminant of the quadratic space Symd(A, σ)0 , sσ is trivial. Moreover, there is a

unique central element · in C1 Symd(A, σ)0 , sσ such that · 2 = 1 and hA (·) = 1.

We may therefore de¬ne a functor

S : C2 ’ B 2

by

S(A, σ) = Symd(A, σ)0 , sσ , · .

(15.16) Theorem. The functors C and S de¬ne an equivalence of groupoids:

B2 ≡ C 2 .

Proof : For any (A, σ) ∈ C2 , the F -algebra homomorphism hA of (??) restricts to

a canonical isomorphism

∼

hA : C0 Symd(A, σ)0 , sσ ’ A,

’

and yields a natural transformation C —¦ S ∼ IdC2 .

=

For (V, q, ζ) ∈ B2 , we de¬ne a linear map mζ : V ’ C0 (V, q) by

mζ (v) = vζ for v ∈ V .

Since s„ (vζ) = (vζ)2 = v 2 = q(v), this map is an isometry:

0

∼

mζ : (V, q) ’ Symd C0 (V, q), „

’ , s„ .

The same argument as in the proof of (??) shows that this isometry carries ζ to the

0

canonical orientation · on Symd C0 (V, q), „ , s„ ; therefore, it de¬nes a natural

transformation S —¦ C ∼ IdB2 which completes the proof.

=

(15.17) Corollary. For every oriented quadratic space (V, q, ζ) of dimension 5,

the Cli¬ord algebra construction yields a group isomorphism

∼

O+ (V, q) = Aut(V, q, ζ) ’ AutF C0 (V, q), „ = PGSp C0 (V, q), „ .

’

Proof : The functor C de¬nes an isomorphism between automorphism groups of

corresponding objects.

Suppose now that (A, σ) ∈ C2 corresponds to (V, q, ζ) ∈ B2 under the equiv-

alence B2 ≡ C2 , so that we may identify (A, σ) = C0 (V, q), „ and (V, q, ζ) =

Symd(A, σ)0 , sσ , · .

(15.18) Proposition. The special Cli¬ord group of (V, q) is

“+ (V, q) = GSp(A, σ).

Under the identi¬cation V = Symd(A, σ)0 ‚ A, the vector representation

χ : GSp(A, σ) ’ O+ (V, q)

is given by χ(g)(v) = gvg ’1 for g ∈ GSp(A, σ) and v ∈ V .

218 IV. ALGEBRAS OF DEGREE FOUR

The spin group is Spin(V, q) = Sp(A, σ) and the group of spinor norms is

Sn(V, q) = G(A, σ).

Proof : By de¬nition, “+ (V, q) is a subgroup of A— , and it consists of similitudes

of (A, σ). We have a commutative diagram with exact rows:

χ

O+ (V, q)

1 ’ ’ ’ F — ’ ’ ’ “+ (V, q) ’ ’ ’

’’ ’’ ’’ ’’’ 1

’’

¦ ¦

¦ ¦

C

1 ’ ’ ’ F — ’ ’ ’ GSp(A, σ) ’ ’ ’ PGSp(A, σ) ’ ’ ’ 1

’’ ’’ ’’ ’’

(see (??)24 ). The corollary above shows that the right-hand vertical map is an

isomorphism, hence “+ (V, q) = GSp(A, σ).

For g ∈ “+ (V, q), we have χ(g)(v) = g · v · g ’1 in C(V, q), by the de¬nition of

the vector representation. Under the identi¬cation V = Symd(A, σ)0 , every vector

v ∈ V is mapped to vζ ∈ A, hence the action of χ(g) is by conjugation by g, since

ζ is central in C(V, q). The last assertions are clear.

An alternate proof is given in (??) below.

If char F = 2, we may combine (??) with the equivalence B2 ≡ B2 of (??) to

get the following relation between groupoids of algebras with involution:

(15.19) Corollary. Suppose char F = 2. The functor S : C2 ’ B2 , which maps

every central simple algebra of degree 4 with symplectic involution (A, σ) to the

algebra of degree 5 with orthogonal involution EndF Symd(A, σ)0 , σsσ where σsσ

is the adjoint involution with respect to sσ , and the functor C : B2 ’ C2 , which

maps every central simple algebra of degree 5 with orthogonal involution (A , σ ) to

its Cli¬ord algebra C(A , σ ), σ , de¬ne an equivalence of groupoids:

C2 ≡ B 2 .

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 )