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A (K1 , a1 )F — (K2 , a2 )F .
Hint: Re¬ne the argument used in the proof of Proposition (??).
6. Let σ, „ be distinct symplectic involutions on a biquaternion F -algebra A. Show
that dimF Symd(A, σ)©Symd(A, „ ) = 2. Show also that there is a quaternion
algebra B ‚ A over some quadratic extension of F such that σ|B = „ |B is the
conjugation involution, and that the algebra B is uniquely determined by this
condition.
7. Let σ, „ , θ be symplectic involutions on a biquaternion F -algebra A. Show that
the invariants of these involutions are related by jσ („ ) = j„ (σ) and jσ („ ) +
j„ (θ) + jθ (σ) ∈ I 3 Wq F . Use this result to show that if σ and „ are conjugate,
then jσ (θ) = j„ (θ).
8. Let σ be a symplectic involution on a biquaternion F -algebra A. Let
Symd(A, σ)0 = { x ∈ Symd(A, σ) | Trpσ (x) = 0 }
and let sσ : Symd(A, σ)0 ’ F be the squaring map. Show that ind(A) ¤ 2 if
and only if sσ is a subform of some (uniquely determined) 3-fold P¬ster form
πσ . Suppose these conditions hold; then
(a) show that (A, σ) has a decomposition
(A, σ) = M2 (F ), σ1 —F (Q, γ)
for some quaternion algebra Q with canonical involution γ and some or-
thogonal involution σ1 on M2 (F ), and that πσ = disc σ1 · NrdQ ;
(b) for θ a hyperbolic involution on A, show that πσ = jθ (σ);
(c) show that G(A, σ) = G(πσ ) = Sn(sσ ).
Hint: The equality G(A, σ) = Sn(sσ ) follows from (??) and G(A, σ) =
G(πσ ) follows from (??) and (??).
9. Suppose char F = 2. Let K/F be an ´tale quadratic extension with non-
e

trivial automorphism ι and let δ ∈ K be such that ι(δ) = ’δ. Let (V, q)
be an odd-dimensional quadratic space over K with trivial discriminant and
let ζ ∈ C(V, q) be an orientation of (V, q). De¬ne an F -linear map i : V ’
M2 NK/F C0 (V, q) by
’δ ι (x · ζ) — 1 ’ 1 — (x · ζ)
0
i(x) = ι
(x · ζ) — 1 + 1 — (x · ζ) 0
for x ∈ V . Show that the map i induces an F -algebra isomorphism:

i— : C V, (trK/F )— ( δ · q) ’ M2 NK/F C0 (V, q) .

Use this result to give a direct proof of the fact that if Q is a quaternion K-
algebra and s : Q0 ’ K is the squaring map on the space of pure quaternions,
then Q0 , (trK/F )— ( δ · s) is an Albert quadratic space of NK/F (Q).
10. Suppose char F = 2. Let Q1 , Q2 be quaternion F -algebras with canonical in-
volutions γ1 , γ2 and let (A, θ) = (Q1 , γ1 ) —F (Q2 , γ2 ). De¬ne a linear endomor-
phism p on Skew(A, θ) = (Q0 —1)•(1—Q0) by p(x1 —1+1—x2) = x1 —1’1—x2
1 2
and a quadratic form q : Skew(A, θ) ’ F by q(x) = xp(x). Consider another
pair of quaternion F -algebras Q1 , Q2 and (A , θ ) = (Q1 , γ1 ) —F (Q2 , γ2 ), and
de¬ne p , q on Skew(A , θ ) as p, q were de¬ned on Skew(A, θ). Show that

for every isomorphism f : (A, θ) ’ (A , θ ) there exists some » ∈ F — such

272 IV. ALGEBRAS OF DEGREE FOUR


that f ’1 —¦ p —¦ f = »p, hence f restricts to a similitude (Skew(A, θ), q) ’ ’
Skew(A , θ ), q with multiplier ».
Hint: Use (??).
This exercise is inspired by Knus-Parimala-Sridharan [?, Theorem 3.4] and
Wadsworth [?]. It shows that the forms q and q are similar without using the
fact that they are Albert forms of A and A .
11. Let σ be a symplectic involution on a biquaternion algebra A. Show that
the invariant jσ of symplectic involutions and the homomorphism ±σ : “ ’
I 3 Wq F/I 4 Wq F are related by
±σ Nrpσ (v), v = jσ Int(v) —¦ σ ’ Nrpσ (v) · Nrpσ +I 4 Wq F
for all v ∈ Symd(A, σ)— .
12. Suppose char F = 2. Let A be a biquaternion division algebra and let x ∈
SL1 (A). √
(a) Let σ be an arbitrary symplectic involution and let L = F ( a) be a
quadratic extension of F in Sym(A, σ) which contains σ(x)x. Recall from
(??) that one can ¬nd b, c, d ∈ F — such that A (a, b)F — (c, d)F . Show
that L contains an element y such that σ(x)x = yy ’1 where is the
nontrivial automorphism of L/F , and that
±(x) = NL/F (y), b, c, d + I 5 F.
(b) Suppose x is contained in a maximal sub¬eld E ‚ A which contains an

intermediate quadratic extension L = F ( a). Recall from (??) that A
(a, b)F — (c, d)F for some b, c, d ∈ F — . Show that L contains an element y
such that NE/L (x) = yy ’1 where is the nontrivial automorphism of L/F ,
and that
±(x) = NL/F (y), b, c, d + I 5 F.
13. Let Q be a quaternion F -algebra. Show that every element in SL1 (Q) is a
commutator, except if F = F2 (the ¬eld with two elements).
Hint: Argue as in (??). If q ∈ SL1 (Q) and 1 + q is not invertible and

nonzero, then show that there is an isomorphism Q ’ M2 (F ) which identi¬es

q with ’1 ’1 . Then q = xyx’1 y ’1 with x = 1 ’1 and y = ’1/2 1 if
1 0
0
0 ’1 0
»+1 0 1 » if char F = 2 and » = 0, 1.
char F = 2, with x = 0 » and y = 0 1
14. Let Q be a quaternion division F -algebra and let (V, h) be a nonsingular her-
mitian space of dimension 2 over Q with respect to the canonical involution,
so that EndQ (V ), σh is a central simple algebra of degree 4 with symplectic
involution. Let g ∈ Sp(V, h) be such that IdV + g is not invertible, and let
v1 ∈ V be a nonzero vector such that g(v1 ) = ’v1 .
(a) If v1 is anisotropic, show that for each vector v2 ∈ V which is orthogonal
to v1 , there is some ± ∈ Q such that g(v2 ) = v2 ± and NrdQ (±) = 1.
(b) If v1 is isotropic, show that for each isotropic vector v2 ∈ V such that
h(v1 , v2 ) = 1, there is some β ∈ Q such that g(v2 ) = v1 β ’ v2 and
TrdQ (β) = 0.
15. Let b be a nonsingular alternating bilinear form on a 4-dimensional F -vector
space V and let g ∈ Sp(V, b) be such that IdV + g is not invertible.
(a) Show that ker(IdV + g) = im(IdV + g)⊥ .
(b) If the rank of IdV + g is 1 or 2, show that V = V1 • V2 for some 2-
dimensional subspaces V1 , V2 which are preserved under g, hence g can be
EXERCISES 273


represented by the matrix
«  « 
’1 1 0 0 ’1 1 0 0
¬ 0 ’1 0 0· ¬ 0 ’1 0 0·
g1 = ¬ · or g2 = ¬ ·.
0 0 ’1 0  0 0 ’1 1 
0 0 0 ’1 0 0 0 ’1
Use Exercise ?? to conclude that g is a commutator if F = F2 . If char F =
2, show that g1 = a1 b1 a’1 b’1 and g2 = a2 b2 a’1 b’1 , where
1 1 2 2
«  « 
1010 1000
¬0 1 0 0 · ¬ ·
· , b1 = ¬0 1 0 0· ,
a1 = ¬0 0 1 0  0 1 1 0
0001 0001
«  « 
0010 1000
¬0 0 0 1· ¬ ·
· , b2 = ¬0 1 0 0· .
a2 = ¬
1 0 0 0 0 0 1 1
0100 0001
(c) If IdV + g has rank 3, show that g can be represented by the matrix
« 
’1 1 0 0
¬ 0 ’1 1 0·
¬ ·
0 1
0 ’1
0 0 0 ’1
(with respect to a suitable basis).
If char F = 2, let
« 
’3 ’8 22 ’12
¬’3 ’9 23 ’12·
y=¬ ·.
’3 ’9 22 ’11
’4 ’5 16 ’8
Show that (x’1)3 (x+1) is the minimum and the characteristic polynomial
of both matrices gy and y. Conclude that there is an invertible matrix x
such that gy = xyx’1 , hence g is a commutator. If char F = 2, let
« 
0100
¬1 0 0 1·
z=¬ ·
1 0 1 1 .
1110
Show that x4 + x3 + x2 + x + 1 is the minimum and the characteristic
polynomial of both matrices gz and z. As in the preceding case, conclude
that g is a commutator.
16. Let A be a biquaternion F -algebra with symplectic involution σ. Write simply
V = Symd(A, σ), V 0 = Symd(A, σ)0 and q = Nrpσ , and let q 0 be the restriction
of q to V 0 . Using (??), show that there is a canonical isomorphism SL1 (A)
Spin(V, q) which maps the subgroup [A— , A— ] to the subgroup
Spin (V, q) = Spin(V 0 , q 0 ) · [„¦, “+ (V 0 , q 0 )]
where „¦ = „¦ EndF (V ), σq , fq is the extended Cli¬ord group. Deduce that
the subgroup Spin (V, q) ‚ Spin(V, q) is generated by the subgroups Spin(U )
for all the proper nonsingular subspaces U ‚ V .
274 IV. ALGEBRAS OF DEGREE FOUR


Hint: Use the proof of (??).
17. Suppose char F = 2 and F contains a primitive 4th root of unity ζ. Let A =
(a, b)F — (c, d)F be a biquaternion division F -algebra.
(a) Show that if A is cyclic (i.e., if A contains a maximal sub¬eld which is
cyclic over F ), then a, b, c, d is hyperbolic.
(b) (Morandi-Sethuraman [?, Proposition 7.3]) Suppose d is an indeterminate
over some sub¬eld F0 containing a, b, c, and F = F0 (d). Show that A is
cyclic if and only if a, b, c is hyperbolic.
Hint: If a, b, c is hyperbolic, the following equation has a nontrivial
solution in F0 :
a(x2 + cx2 ) + b(y1 + cy2 ) ’ ab(z1 + cz2 ) = 0.
2 2 2 2
1 2
√ √ √√
Let e = 2 c(ax1 x2 + by1 y2 ’ abz1 z2 ) ∈ F0 ( c). Show that F0 ( c)( e) is
cyclic over F0 and splits (a, b)F (√c) , hence also A.



Notes
§??. If char F = 2, an alternative way to de¬ne the canonical isomorphism
N —¦ C(A, σ) (A, σ) for (A, σ) ∈ D2 (in (??)) is to re¬ne the fundamental re-
lation (??) by taking the involutions into account. As pointed out in the notes
for Chapter ??, one can de¬ne a nonsingular hermitian form H on the left A-
submodule B (A, σ) ‚ B(A, σ) of invariant elements under the canonical involution
ω and show that the canonical isomorphism ν of (??) restricts to an isomorphism
of algebras with involution:

NZ/F C(A, σ), σ ’ EndA B (A, σ), σH

where Z is the center of the Cli¬ord algebra C(A, σ). Since deg A = 4, dimension
count shows that the canonical map b : A ’ B(A, σ) induces an isomorphism of A-

modules A ’ B (A, σ). Moreover, under this isomorphism, H (a1 , a2 ) = 2a1 σ(a2 )

for a1 , a2 ∈ A. Therefore, b induces a canonical isomorphism

(A, σ) ’ EndA B (A, σ), σH .

Similarly, in Theorem (??) the canonical isomorphism D —¦ C(A, σ) (A, σ) for
(A, σ) ∈ D3 can be derived from properties of the bimodule B(A, σ). De¬ne a left
C(A, σ) —Z C(A, σ)-module structure on B(A, σ) by
(c1 — c2 ) u = c1 — u · σ(c2 ) for c1 , c2 ∈ C(A, σ), u ∈ B(A, σ).
If g ∈ C(A, σ) —Z C(A, σ) is the Goldman element of the Cli¬ord algebra, the map
C(A, σ) —Z C(A, σ) ’ B(A, σ) which carries ξ to ξ 1b induces an isomorphism of
left C(A, σ) —Z C(A, σ)-modules

[C(A, σ) —Z C(A, σ)](1 ’ g) ’ B(A, σ).

We thus get a canonical isomorphism
»2 C(A, σ) EndC(A,σ)—Z C(A,σ) B(A, σ).
Using (??), we may identify the right-hand side with A —F Z and use this identi¬-
cation to get a canonical isomorphism D C(A, σ), σ A.
In the proofs of Theorems (??) and (??), it is not really necessary to consider
the split cases separately if char F = 0; one can instead use results on the extension
NOTES 275


of Lie algebra isomorphisms in Jacobson [?, Ch. 10, §4] to see directly that the
™1 1 ™™ 1 ™
1
canonical Lie algebra isomorphisms n—¦ 2 c, 2 c—¦n, »2 —¦ 2 c and 2 c—¦»2 extend (uniquely)
to isomorphisms of the corresponding algebras with involution (see Remarks (??)
and (??)).
The fact that a central simple algebra of degree 4 with orthogonal involution
decomposes into a tensor product of stable quaternion subalgebras if and only if
the discriminant of the involution is trivial (Corollary (??) and Proposition (??))
was proved by Knus-Parimala-Sridharan [?, Theorem 5.2], [?]. (The paper [?] deals
with Azumaya algebras over rings in which 2 is invertible, whereas [?] focuses on
central simple algebras, including the characteristic 2 case).
The description of groups of similitudes of quadratic spaces of dimension 3, 4,
5 and 6 dates back to Van der Waerden [?] and Dieudonn´ [?, §3]. The case of
e
quadratic spaces over rings was treated by Knus [?, §3] and by Knus-Parimala-
Sridharan [?, §6] (see also Knus [?, Ch. 5]). Cli¬ord groups of algebras of degree 4
or 6 with orthogonal involution are determined in Merkurjev-Tignol [?, 1.4.2, 1.4.3].
§??. Albert forms are introduced in Albert [?]. Theorem 3 of that paper yields
the criterion for the biquaternion algebra A = (a, b)F — (c, d)F to be a division alge-
bra in terms of the associated quadratic form q = a, b, ’ab, ’c, ’d, cd . (See (??).)
The de¬nition of the form q thus depends on a particular decomposition of the bi-
quaternion algebra A. The fact that quadratic forms associated to di¬erent decom-
positions are similar was ¬rst proved by Jacobson [?, Theorem 3.12] using Jordan
structures, and later by Knus [?, Proposition 1.14] and Mammone-Shapiro [?] us-
ing the algebraic theory of quadratic forms. (See also Knus-Parimala-Sridharan [?,
Theorem 4.2].) Other proofs of Albert™s Theorem (??) were given by P¬ster [?,
p. 124] and Tamagawa [?] (see also Seligman [?, App. C]). A notion of Albert form
in characteristic 2 is discussed in Mammone-Shapiro [?]. Note that the original
version of Jacobson™s paper [?] does not cover the characteristic 2 case adequately;
see the reprinted version in Jacobson™s Collected Mathematical Papers [?], where
the characteristic is assumed to be di¬erent from 2.
From the de¬nition of the quadratic form q = a, b, ’ab, ’c, ’d, cd associated
to A = (a, b)F — (c, d)F , it is clear that q is isotropic if and only if the quaternion
algebras (a, b)F and (c, d)F have a common maximal sub¬eld. Thus Corollary (??)
easily follows from Albert™s Theorem (??). The proof given by Albert in [?] is more
direct and also holds in characteristic 2. Another proof (also valid in characteris-
tic 2) was given by Sah [?]. If the characteristic is 2, the result can be made more
precise: if a tensor product of two quaternion division algebras is not a division
algebra, then the two quaternion algebras have a common maximal sub¬eld which
is a separable extension of the center. This was ¬rst shown by Draxl [?]. The proofs
given by Knus [?] and by Tits [?] work in all characteristics, and yield the more
precise result in characteristic 2.
The fact that G(q) = F —2 · NrdA (A— ) for an Albert form q of a biquaternion
algebra A is already implicit in Van der Waerden [?, pp. 21“22] and Dieudonn´ [?, e
Nos 28, 30, 34]. Knus-Lam-Shapiro-Tignol [?] gives other characterizations of that
group. In particular, it is shown that this group is also the set of discriminants of
orthogonal involutions on A; see Parimala-Sridharan-Suresh [?] for another proof
of that result.
If σ is a symplectic involution on a biquaternion algebra A, the (Albert) quad-
ratic form Nrpσ de¬ned on the space Symd(A, σ) is the generic norm of Symd(A, σ),
viewed as a Jordan algebra, see Jacobson [?]. This is the point of view from which
276 IV. ALGEBRAS OF DEGREE FOUR


results on Albert forms are derived in Jacobson [?, Ch. 6, §4]. The invariant jσ („ )
of symplectic involutions is de¬ned in a slightly di¬erent fashion in Knus-Lam-
Shapiro-Tignol [?, §3]. See (??) for the relation between jσ („ ) and Rost™s higher
cohomological invariants.
If σ is an orthogonal involution on a biquaternion algebra A, the linear endomor-
phism pσ of Skew(A, σ) was ¬rst de¬ned by Knus-Parimala-Sridharan [?], [?] (see
also Knus [?, Ch. 5]), who called it the pfa¬an adjoint because of its relation with
the pfa¬an. Pfa¬an adjoints for algebras of degree greater than 4 are considered
in Knus-Parimala-Sridharan [?]. If deg A = 2m, the pfa¬an adjoint is a polynomial
map of degree m ’ 1 from Skew(A, σ) to itself. Knus-Parimala-Sridharan actually
treat orthogonal and symplectic involutions simultaneously (and in the context of
Azumaya algebras): if σ is a symplectic involution on a biquaternion algebra A,
the pfa¬an adjoint is the endomorphism of Sym(A, σ) de¬ned in §??.
Further results on Albert forms can be found in Lam-Leep-Tignol [?], where
maximal sub¬elds of a biquaternion algebra are investigated; in particular, neces-
sary and su¬cient conditions for the cyclicity of a biquaternion algebra are given
in that paper.
§??. Suppose char F = 2. As observed in the proof of (??), the group “ =
{ (», a) ∈ F — —A— | »2 = NrdA (a) }, for A a biquaternion F -algebra, can be viewed
as the special Cli¬ord group of the quadratic space Sym(A, σ), Nrpσ for any
symplectic involution σ. The map ±σ : “ ’ I 4 F/I 5 F can actually be de¬ned on the
full Cli¬ord group “ Sym(A, σ), Nrpσ by mapping the generators v ∈ Sym(A, σ)—
to ¦v + I 5 F . Showing that this map is well-de¬ned is the main di¬culty of this
approach.
The homomorphism ± : SK1 (A) ’ I 4 F/I 5 F (for A a biquaternion algebra) was
originally de¬ned by Rost in terms of Galois cohomology, as a map ± : SK1 (A) ’
H 4 (F, µ2 ). Rost also proved exactness of the following sequence:
± h
0 ’ SK1 (A) ’ H 4 (F, µ2 ) ’ H 4 (F (q), µ2 )
’ ’
where h is induced by scalar extension to the function ¬eld of an Albert form
q, see Merkurjev [?, Theorem 4] and Kahn-Rost-Sujatha [?]. The point of view
of quadratic forms developed in §?? is equivalent, in view of the isomorphism

e4 : I 4 F/I 5 F ’ H 4 (F, µ2 ) proved by Rost (unpublished) and more recently by

Voevodsky, which leads to a commutative diagram
± i
0 ’ ’ ’ SK1 (A) ’ ’ ’ I 4 F/I 5 F ’ ’ ’ I 4 F (q)/I 5 F (q)
’’ ’’ ’’
¦ ¦
¦ ¦
e e
4 4


± h
0 ’ ’ ’ SK1 (A) ’ ’ ’ H 4 (F, µ2 ) ’ ’ ’ H 4 F (q), µ2 .
’’ ’’ ’’
The fact that the sequences above are zero-sequences readily follows from functo-
riality of ± and ± , since SK1 (A) = 0 if A is not a division algebra or, equivalently
by (??), if q is isotropic. In order to derive exactness of the sequence above from
exactness of the sequence below, only “elementary” information on the map e 4 is
needed: it is su¬cient to use the fact that on P¬ster forms e4 is well-de¬ned (see
Elman-Lam [?, 3.2]) and injective (see Arason-Elman-Jacob [?, Theorem 1]). In
fact, (??) shows that the image of ± consists of 4-fold P¬ster forms (modulo I 5 F )
and, on the other hand, the following proposition also shows that every element in
ker i is represented by a single 4-fold P¬ster form:
NOTES 277


(17.30) Proposition. If q is an Albert form which represents 1, then
ker i = { π + I 5 F | π is a 4-fold P¬ster form containing q }.
Proof : By Fitzgerald [?, Corollary 2.3], the kernel of the scalar extension map
W F ’ W F (q) is an ideal generated by 4-fold and 5-fold P¬ster forms. Therefore,
every element in ker i is represented by a sum of 4-fold P¬ster forms which become
hyperbolic over F (q). By the Cassels-P¬ster subform theorem (see Scharlau [?,
Theorem 4.5.4]) the P¬ster forms which satisfy this condition contain a subform
isometric to q. If π1 , π2 are two such 4-fold P¬ster forms, then the Witt index of
π1 ⊥ ’π2 is at least dim q = 6. By Elman-Lam [?, Theorem 4.5] it follows that
and elements a1 , a2 ∈ F — such that
there exists a 3-fold P¬ster form
π1 = · a1 and π2 = · a2 .
Therefore,
mod I 5 F.
π1 + π 2 ≡ · a 1 a2
By induction on the number of terms, it follows that every sum of 4-fold P¬ster
forms representing an element of ker i is equivalent modulo I 5 F to a single 4-fold
P¬ster form.
The image of the map ±σ can be described in a similar way. Since ±σ = 0 if σ
is hyperbolic or, equivalently by (??), if the 5-dimensional form sσ is isotropic, it
follows by functoriality of ±σ that im ±σ lies in the kernel of the scalar extension
map to F (sσ ). One can use the arguments in the proof of Merkurjev [?, Theorem 4]
to show that this inclusion is an equality, so that the following sequence is exact:
±
“σ ’’ I 4 F/I 5 F ’’ I 4 F (sσ )/I 5 F (sσ ).
σ

Corollary (??) shows that every element in [A— , A— ] is a product of two com-
mutators. If A is split, every element in [A— , A— ] can actually be written as a single
commutator, as was shown by Thompson [?]. On the other hand, Kursov [?] has
found an example of a biquaternion algebra A such that the group [A— , A— ] does
not consist of commutators, hence our lower bound for the number of factors is
sharp, in general.
The ¬rst example of a biquaternion algebra A such that SK1 (A) = 0 is due
to Platonov [?]. For a slightly di¬erent relation between the reduced Whitehead
group of division algebras and Galois cohomology of degree 4, see Suslin [?].
Along with the group Σσ (A), the group generated by skew-symmetric units in a
central simple algebra with involution (A, σ) is also discussed in Yanchevski˜ [?]. If ±

σ is orthogonal, it is not known whether this subgroup is normal in A . Triviality of
the group USK1 (B, „ ) can be shown not only for division algebras of prime degree,
but also for division algebras whose degree is square-free; indeed, the exponent of
USK1 (B, „ ) divides deg B/p1 . . . pr , where p1 , . . . , pr are the prime factors of deg B:
see Yanchevski˜ [?]. Examples where the group K1 Spin(A) is not trivial are given
±
in Monastyrny˜ ±-Yanchevski˜ [?]. See also Yanchevski˜ [?] for the relation between
± ±
K1 Spin and decomposability of involutions.
278 IV. ALGEBRAS OF DEGREE FOUR
CHAPTER V


Algebras of Degree Three

The main topic of this chapter is central simple algebras of degree 3 with
involutions of the second kind and their ´tale (commutative) subalgebras. To every
e
involution of the second kind on a central simple algebra of degree 3, we attach
a 3-fold P¬ster form, and we show that this quadratic form classi¬es involutions
up to conjugation. Involutions whose associated P¬ster form is hyperbolic form
a distinguished conjugacy class; we show that such an involution is present on
every central simple algebra of degree 3 with involution of the second kind, and we
characterize distinguished involutions in terms of ´tale subalgebras of symmetric

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