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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
equation in ±
qσ (x + ±y) = qσ (x) + ±qσ (x, y) + ±2 qσ (y) = 0
has at most two solutions and F has more than three elements. For this choice
of ±, we have
qσ (x + ±y)
pσ (x + ±y) = pσ (x + ±y).
qσ (x + ±y)
By linearity of pσ and pσ , it follows that
qσ (x + ±y) q (x + ±y)
pσ (x) + ± σ
pσ (x) + ±pσ (y) = pσ (y).
qσ (x + ±y) qσ (x + ±y)
§16. BIQUATERNION ALGEBRAS 247


On the other hand, we also have
qσ (x) q (y)
pσ (x) + ± σ
pσ (x) + ±pσ (y) = pσ (y),
qσ (x) qσ (y)
hence
qσ (x) q (x + ±y) q (y)
=σ =σ
qσ (x) qσ (x + ±y) qσ (y)
since pσ (x) and pσ (y) are linearly independent.
To conclude, observe that Skew(A, σ) has a basis (ei )1¤i¤6 consisting of in-
vertible elements: this is clear if σ is the involution θ de¬ned above, and it follows
for arbitrary σ since Skew(A, σ) = v · Skew(A, θ) if σ = Int(v) —¦ θ. Denoting
» = qσ (e1 )qσ (e1 )’1 , the argument above shows that
qσ (ei )

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