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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 )

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