Proposition 4

Let $\beta$ be the principal antiautomorphism of $C(Q)$ (n° 1). For every $s \in G^+$, $\beta(s)s$ is a scalar. The mapping $N : s \to \beta(s)s$ is a homomorphism of $G^+$ into the multiplicative group $A^*$ of nonzero elements of $A$.

Indeed, for $s \in G^+$, one has $sEs^{-1} = E$, whence $\beta(s)^{-1}E\beta(s) = E$, which shows that $\beta(s) \in G^+$. Since $sx = (\varphi(s).x)s$ for all $x \in E$, one has $x\beta(s) = \beta(sx) = \beta(s)(\varphi(s).x)$, and consequently $\beta(s)sx = \beta(s)(\varphi(s).x)s = x\beta(s)s$, which implies that $\beta(s)s$ belongs to the center of $C(Q)$. Since, moreover, $\beta(s)s$ belongs to $C^+(Q)$, $\beta(s)s$ is a scalar (Theorem 2 and 3). Finally one has $\beta(st)st = \beta(t)\beta(s)st = \beta(s)s\beta(t)t$, that is to say $N(st) = N(s)N(t)$ for $s, t$ in $G^+$. QED.

The scalar $N(s) = \beta(s)s$ ($s \in G^+$) is called the spinor norm of $s$. We denote by $G_0^+$ and call the reduced Clifford group the kernel of the homomorphism $N$. The image $\varphi(G_0^+)$ is denoted by $O_0^+(Q)$ and is called the reduced orthogonal group of $Q$. Since the kernel of the restriction of $\varphi$ to $G^+$ is the set of even and invertible elements of the center of $C(Q)$ (Theorem 4) and is therefore identified with $A^*$ (Theorem 2 and 3), $\varphi(G^+)/O_0^+(Q)$ is isomorphic to $G^+/A^*G_0^+$, and hence also to $N(G^+)/N(A^*)$, and in particular commutative. It is clear that $N(A^*)$ is the subgroup $(A^*)^2$ of squares of elements of $A^*$. If $Q$ is of index $> 0$, there exist, whatever $a \in A^*$ may be, two elements $x$ and $y$ of $E$ such that $Q(x) = a$ and $Q(y) = 1$ (§ 4, No. 2, Proposition 4) ; since $xy \in G^+$ and $N(xy) = Q(x)Q(y) = a$, this shows that $N(G^+) = A^*$, and hence that $\varphi(G^+)/O_0^+(Q)$ is isomorphic to $A^*/(A^*)^2$.

Exercices. — ¶ 1) Prove Cor. 3 and 4 of Theorem 1 of No. 3 when $E_1$ and $E_2$ are any two supplementary submodules in $E$. (First establish Cor. 4: now prove for this purpose that the tensor product $C(Q_1) \otimes C(Q_2)$ is endowed with an algebra structure by the sign convention made in the statement of Cor. 4, and that this algebra $S$ is solution of the same universal mapping problem as $C(Q)$; for this purpose, one will consider, for every linear mapping $f$ of $E$ into an algebra $D$ over $A$ such that $(f(x))^2 = Q(x).1$, the homomorphism $\bar{f}_i$ of $C(Q_i)$ into $D$ such that $\bar{f}_i(1) = 1$, $\bar{f}_i(\rho_{Q_i}(x_i)) = f(x_i)$ for $x_i \in E_i$ ($i = 1, 2$) and one will prove that there exists a homomorphism $\bar{f}$ of the algebra $S$ into $D$ such that

for $z_i \in C(Q_i)$ ($i = 1, 2$). To prove then Cor. 3, consider the quadratic form $Q'$ external direct sum of $Q_1$ and $Q_2$ (§ 3, No. 4), and remark that one has $Q'(x) = Q(x) + F(x, x)$, $F$ being the bilinear form defined by $F(x_1 + x_2, y_1 + y_2) = -\Phi(x_1, y_2)$ for $x_i, y_i$ in $E_i$ ($i = 1, 2$) ; use then Prop. 3.)

¶ 2) Suppose that $E$ be direct sum of two orthogonal submodules $E_1, E_2$ and denote by $Q_i$ the restriction of $Q$ to $E_i$ ($i = 1, 2$);

on suppose en outre qu’il existe $u \in C^+(Q_2)$ tel que $u^2 = a . 1$, $a$ étant invér sible dans $A$, et que $u \rho_{Q_2}(x_2) = -\rho_{Q_2}(x_2)$ pour tout $x_2 \in E_2$. Now prove qu’il existe un isomorphisme $\varphi$ de $C(Q)$ sur le produit tensoriel $C(aQ_1) \otimes C(Q_2)$ (tel qu’il est défini dans le chap. III, § 3, n° 1), ayant la propriété que pour tout $x = x_1 + x_2$ ($x_i \in E_i, i = 1, 2$), on a

(Proving the existence of the homomorphism $\varphi$ as a consequence of the universal property of $C(Q)$. On the other hand, there is a homomorphism $g_1$ of $C(aQ_1)$ into $C(Q)$ such that $g_1(\rho_{aQ_1}(x_1)) = h_2(u)\rho_{Q_1}(x_1)$, where $h_2$ denotes the canonical homomorphism of $C(Q_2)$ into $C(Q)$; remark then that $g_1(z_1)$ and $h_2(z_2)$ commute for $z_1 \in C(aQ_1)$ and $z_2 \in C(Q_2)$, and deduce the existence of a reciprocal homomorphism to $\varphi$.)

In what case does this result apply when $A$ is a field of characteristic $\neq 2$ (use Remark 2 following Cor. of Th. 2)?

3) a) With the notation of No. 2, let $x_i$ ($1 \leqslant i \leqslant n$) be elements of $E$ such that $f(x_i) = 0$; show that we have $i_f(x_1 \otimes x_2 \otimes \cdots \otimes x_n) = 0$. In particular, if $F$ is a bilinear form on $E$ such that $F(x_i, x_j) = 0$ for $i > j$, we have $i_F^E(x_2 \otimes x_3 \otimes \cdots \otimes x_n) = 0$.

b) With the notation of Prop. 3 of No. 3, let $x_i$ ($1 \leqslant i \leqslant n$) be elements of $E$ such that $F(x_i, x_j) = 0$ for $i > j$. Show that we have $\overline{\lambda}_F(\rho_{Q'}(x_1) \ldots \rho_{Q'}(x_n)) = \rho_Q(x_1) \ldots \rho_Q(x_n)$ (use a) and formula (10) of No. 2).

c) We suppose that $A$ is a field of characteristic $\neq 2$; for every quadratic form $Q$ on $E$, let $\mu_Q$ be the mapping $\overline{\lambda}_F$ of $C(Q)$ onto $\wedge E$ corresponding to $F(x, y) = \frac{1}{2} \Phi(x, y)$. Now prove that if the vectors $x_i$ ($1 \leqslant i \leqslant n$) are pairwise orthogonal, one has

Deduce that if $y_i$ ($1 \leqslant i \leqslant n$) are $n$ arbitrary vectors, one has

4) Let $C_h$ be the submodule of $C(Q)$ generated by the products of $k$ elements of $\rho_Q(E)$ for $0 \leqslant k \leqslant h$. Show that the mapping

defines, on passing to the quotients, an alternating multilinear mapping of $E^h$ into $C_h/C_{h-1}$. One deduces from this a linear mapping $\pi_h$ of $\wedge^h E$ into $C_h/C_{h-1}$; show that $\pi_h$ is an isomorphism when $E$ is a free module. If $f$ is an orthogonal transform, one has $C(f) \circ \pi_h = \pi_h \circ (\wedge^h f)$. When $A$ is a field of characteristic $\neq 2$, show that $\pi_h$ is the restriction of the inverse mapping $\mu_Q^{-1}$ defined in exerc. 3 c).

5) With the notation of No. 2, consider the mapping $i_f$ of $\wedge E$ into itself. Show that one has

and deduce that $i_f$ is none other than the right interior product $z \to z \llcorner f$ (chap. III, § 8, No. 4, def. 2).

6) Show that, if $E$ admits an orthogonal basis $(e_1, e_2)$ for Q, and if we put $\alpha_i = Q(e_i)$ ($i = 1, 2$), the algebra $C(Q)$ is isomorphic to the quaternion algebra over $A$ corresponding to the pair $(\alpha_1, \alpha_2)$.

7) Let $A$ be a maximal ordered field, $E$ a vector space of even dimension $2r$ over $A$, and $Q$ a non-degenerate quadratic form on $E$, positive or negative. Show that, if $Q$ is positive, $C(Q)$ is isomorphic to a matrix algebra over $A$ when $r(r-1)/2$ is even, and to a matrix algebra over the quaternion algebra over $A$ when $r(r-1)/2$ is odd. If $Q$ is negative, $C(Q)$ is isomorphic to a matrix algebra over $A$ when $r(r+1)/2$ is even, and to a matrix algebra over the quaternion algebra over $A$ when $r(r+1)/2$ is odd (use exerc. 2 and 6). What is the structure of $C^+(Q)$ in these different cases?

8) Let $A$ be the ring $\mathbf{Z}/(4)$, $E$ the $A$-module $\mathbf{Z}/(2)$; if we put Q(0) = 0, Q(u) = 1 for the unique element $u \neq 0$ of $E$, $Q$ is a quadratic form on $E$. Show that the $A$-modules $C(Q)$ and $\wedge E$ are not isomorphic (it will be proved that $C(Q)$ is isomorphic to the direct sum of two modules isomorphic to E).

¶ 9) We suppose that $A$ is a field of characteristic 2, $E$ a vector space of finite dimension $2r$, $Q$ a non-degenerate quadratic form on $E$.

a) Let $(e_i)$ be a symplectic basis ($§ 5,$ No. 1) of $E$ for the alternating bilinear form $\Phi$ associated with Q. Show that the element

of $C^+(Q)$ forms, together with the unit element, a basis of the center Z of $C^+(Q)$. In order that Z be the direct sum of two fields, it is necessary and sufficient that the element

(called the pseudo-discriminant of Q with respect to the symplectic basis $(e_i)$) be of the form $\lambda^2 + \lambda$, with $\lambda \in A$.

b) Let $u$ be a similarity for $\Phi$ ($§ 6,$ No. 5) with multiplier $\mu(u)$, and let $Q_1(x) = Q(u(x))$. Put

and $Q(e_{2i-1}) = \alpha_i, Q(e_{2i}) = \beta_i$ ($1 \leq i \leq r$). Now prove that

where (Dickson invariant of $u$ with respect to the basis $(e_i)$). (Notice that the element belongs to $\mathbf{Z}$.) In order that $u$ be a similarity for $Q$ ($§ 4$, Exercise 9) with multiplier $\mu(u)$, it is necessary and sufficient that $\mathrm{D}(u) = 0$ or $\mathrm{D}(u) = \mu(u)$; the similarities for $Q$ such that $\mathrm{D}(u) = 0$ are called direct.

c) Now prove that, if $\varphi$ is a similarity for $\Phi$, $u$ a similarity for $Q$, one has

(consider the Dickson invariant of $\varphi$ with respect to the symplectic basis formed by the $u(e_{2i-1})$ and $(\mu(u))^{-1}u(e_{2i})$ for $1 \leq i \leq r$). Deduce that the direct similarities for $Q$ form a normal subgroup of index 2 in the group of similarities for $Q$.

d) If $u$ is the symmetry with respect to the hyperplane orthogonal to a nonsingular vector in $E$, now prove that $\mathrm{D}(u) = 1$. Deduce that the group $\varphi(G^+)$ (notations of No. 5) is the group $\mathbf{SO}(Q)$ defined in Exercise 28 c) of $§ 6$.

e) Let $u \in \mathbf{O}(Q)$, and suppose $\Lambda$ is algebraically closed. Show that, in order that $u \in \mathbf{SO}(Q)$, it is necessary and sufficient that the number of elementary divisors of the module $E_u$ be pair. (With the notations of Exercise 15 of $§ 5$, remark first that if $p, q$ are two distinct irreducible factors of the minimal polynomial of $u$, the number of indecomposable submodules of which $G(p, q)$ is direct sum is equal to the number of indecomposable submodules of which $G(q, p)$ is direct sum. Remark on the other hand that $G(p, p) = \{0\}$ unless for $p = X - 1$. Prove finally that one can restrict oneself to the case where $E_u$ is equal to $G(p, p)$ (with $p = X - 1$) and is indecomposable. There is then a symplectic basis $(e_i)$ of $E$ such that $e_1, e_3, \ldots, e_{2k-1}$ form a basis of $(p(u))^{2r-k}(E)$ for $1 \leq k \leq r$; show that $e_1, e_3, \ldots, e_{2r-3}$ are singular vectors, and conclude that $\mathrm{D}(u) = 1$.)

$§ 10$ We suppose that $A$ is a field, $E$ a finite-dimensional vector space, $Q$ a degenerate quadratic form on $E$; let $M$ be a supplementary subspace of $E^0$ in $E$, $B$ the Clifford algebra (semisimple) of the restriction of $Q$ to $M$.

a) We suppose first that $A$ is of characteristic $\neq 2$. Let $L$ be the Clifford algebra of the restriction of $Q$ to $E^0$ (isomorphic to $\wedge E^0$), $\mathscr{R}_0$ its radical (ideal generated in $L$ by $E^0$, and of codimension 1 in $L$); show that the radical $\mathscr{R}$ of $C(Q)$ is obtained (up to an isomorphism) by defining the algebra structure on $B \otimes_A \mathscr{R}_0$ as in Cor. 4 of Theorem 1, that $C(Q)/\mathscr{R}$ is isomorphic to $B$ and $C(Q)$ is direct sum of $B$ and $\mathscr{R}$.

b) Suppose that A has characteristic 2. Let F be the subspace of E⁰ formed by the singular vectors $x \in E^0$, and let N be a supplementary subspace of F with respect to E⁰. If $(a_i)_{1 \leq i \leq d}$ is a basis of N, and Q$(a_i) = \alpha_i$, the elements $\alpha_i^{1/2}$, in an algebraic closure of A, are linearly independent over A. Let $(\alpha_i^{1/2})_{1 \leq i \leq e}$ be a 2-basis of the field $A_1 = A(\alpha_1^{1/2}, \ldots, \alpha_d^{1/2})$ over A (chap. V, § 8, exerc. 1), and let $h = \dim F$. If B₁ is the central simple algebra $B \otimes_A A_1$, now prove that C(Q) is isomorphic to the algebra $B_1 \otimes_{A_1} L_1$, where $L_1$ is the exterior algebra of a vector space of dimension $h + d - e$ over $A_1$. If $R_1$ is the radical of $L_1$ (of codimension 1 (over $A_1$) in $L_1$), the radical $R$ of C(Q) is isomorphic to the algebra $B_1 \otimes_{A_1} R_1$, $C(Q)/R$ is isomorphic to $B_1$, and C(Q) is a direct sum of $B_1$ and $R$.

c) The hypotheses being those of b), suppose in addition that $h = 0$, $d = 1, e = 0$ (which implies that dim M is even). If $Q_0$ is the restriction of Q to M, now prove that $C^+(Q)$ is isomorphic to $C(Q_0)$.

¶ 11) a) With the hypotheses and notations of No. 5, now prove that $N(G^+)$ is the subgroup of $A^*$ generated by the products $Q(x)Q(y)$, where $x$ and $y$ run through the set of non-isotropic vectors of E. (Reduce to the case $A \neq \mathbf{F}_2$; use prop. 5 and exerc. 28 c) of § 6, as well as exerc. 9 d) of § 9.) Case where Q is of index $\geq 1$.

b) Suppose in addition that $A \neq \mathbf{F}_2$, that E is of dimension $n \geq 2$ and that Q is of index $> 0$. Now prove that $O_0^+(Q)$ is the commutator group of the group $O(Q)$. (Reduce to the case $n = 2$, using exerc. 17 c) and 28 e) of § 6.)

c) Retain the hypotheses of b), and suppose in addition that A has characteristic $\neq 2$ and that E is of even dimension. For the automorphism $x \to -x$ of E to belong to $O_0^+(Q)$ it is necessary and sufficient that the discriminant of Q (with respect to any basis of E) be a square.

¶ 12) a) Let $a$ be an invertible element of A; now prove that there exists one and only one isomorphism $\theta_a$ of $C^+(Q)$ onto $C^+(aQ)$ such that

for all $x, y$ in E ($\rho$ and $\rho_1$ denoting the canonical mappings of E into $C(Q)$ and $C(aQ)$ respectively). Deduce that if A is a field of characteristic $\neq 2$, E a vector space of odd dimension and Q a non-degenerate quadratic form, $C(Q)$ and $C(aQ)$ are isomorphic (cf. exerc. 7).

b) Suppose that $A$ be a field, $E$ a vector space of even dimension $2r > 0$, $Q$ a non-degenerate quadratic form. Let $u$ be a similitude relative to $Q$; prove that there exists one and only one $A$-automorphism $\bar{u}$ of the algebra $C^+(Q)$ such that for $1 \leq h \leq r$ and $x_i \in E$ ($1 \leq i \leq 2h$), one has

where $\mu$ denotes the multiplier of $u$. For $\bar{u}$ to be an inner automorphism, it is necessary and sufficient that $u$ be a direct similitude (§ 6, No. 5 and § 9, Exercise 9 b)). Suppose in addition that $r \geq 2$; then, for $\bar{u}$ to be the identity, it is necessary and sufficient that $u$ be a homothety.

Prove that the automorphism $\bar{u}$ of $C^+(Q)$ is the restriction to $C^+(Q)$ of an inner automorphism of $C(Q)$.

¶ 13) Suppose that $A$ be a field of characteristic $\neq 2$, $E$ a vector space of even dimension $2r > 0$, $Q$ a non-degenerate quadratic form. We denote by $E_h$ the inverse image of $\wedge^h E$ by the isomorphism $\mu_Q$ of Exercise 3 c), so that $E_h$ is the subspace of $C(Q)$ generated by the products $x_1 x_2 \ldots x_h$, where the $x_i$ ($1 \leq i \leq h$) are pairwise orthogonal.

a) Let $\alpha$ be an element of $A$, $s$ an element of $E_2$. Prove that, if $(\alpha + s)^2 \in A$, either $s = 0$, or $\alpha = 0$ and $s = xy$, where $x$ and $y$ are two orthogonal vectors. (Remark that if $t = \mu_Q(\alpha + s)$, $t \wedge t$ necessarily belongs to $A + \wedge^2 E$, by expressing $s$ with the aid of an orthogonal basis of $E$; deduce that one necessarily has $t = \beta + x \wedge y$ with $\beta \in A$, $x, y$ in $E$, using § 5, No. 1, Cor. 2 of Theorem 1).

b) Let $(x, y), (u, v)$ be two pairs of orthogonal non-isotropic vectors in $E$, $P_{xy}$ and $P_{uv}$ the planes $Ax + Ay, Au + Av$ respectively. In order that one have $(xy)(uv) = -(uv)(xy)$ in $C(Q)$, it is necessary and sufficient that $P_{xy} + P_{uv}$ be a non-isotropic subspace of dimension 3, in which $P_{xy}$ and $P_{uv}$ are weakly orthogonal (§ 3, exerc. 11).

c) Let $g$ be an $A$-automorphism of $C^+(Q)$ transforming $E_2$ into itself; prove that there exists a similitude $u$ relative to $Q$ such that $g = \bar{u}$ (exerc. 12 b)). (Let $(e_i)_{1 \leq i \leq 2r}$ be an orthogonal basis of $E$; using a and b), prove that there exists an orthogonal basis $(x_i)$ of $E$ such that $g(e_1 e_i) = x_1 x_i$ for $2 \leq i \leq 2r$.

14) Suppose that $A$ is a field of characteristic $\neq 2$, $E$ a vector space of dimension $n$ over $A$, $Q$ and $Q_1$ two non-degenerate quadratic forms on $E$; let $\tilde{\Delta}$ (resp. $\tilde{\Delta}_1$) be the class of the discriminant $\Delta$ of $Q$ (resp. of the discriminant $\Delta_1$ of $Q_1$) with respect to a basis of $E$, in the group $A^*/(A^*)^2$, a class which does not depend on the basis chosen in $E$. Suppose that $n = 2$.

a) Prove that if $\tilde{\Delta} = \tilde{\Delta}_1$ and if the Clifford algebras $C(Q)$ and $C(Q_1)$ are isomorphic, $Q$ and $Q_1$ are equivalent (consider on $C(Q)$ the quadratic form $z \mapsto z \bar{z}$, where $z \mapsto \bar{z}$ is the unique involutive antiautomorphism of $C(Q)$ whose set of invariants is the center of $C(Q)$ (cf. exerc. 6 and chap. VIII, § 11, exerc. 5 e)); apply the Witt theorem).

(b) In order that $C(Q)$ be isomorphic to the matrix algebra $\mathbf{M}_2(A)$ it is necessary and sufficient that, for at least one $x \neq 0$ in $E$, there exist $y, z$ in $E$ such that $Q(x) + Q(y)Q(z) = 0$; if this is so, this property is true for every $x \neq 0$ in $E$.

¶ 15) We retain the general notation and hypotheses of Exercise 14, but suppose $n = 3$.

(a) Now prove that $C^+(Q)$ is isomorphic to a quaternion algebra over $A$ and that $\beta$ is the antiautomorphism $z \mapsto \bar{z}$ of this algebra whose set of invariants is the center of $C^+(Q)$; if $P$ is the subspace of $C^+(Q)$ formed by the pure quaternions (that is, such that $z = -\bar{z}$; chap. VIII, § 11, Exercise 6), the restriction to $P$ of the quadratic form $z \mapsto z \bar{z}$ is equivalent to $\lambda Q$, where $\lambda \in A$. Deduce that, in order that $C^+(Q)$ be a field, it is necessary and sufficient that $Q$ have index 0.

(b) Now prove that if $\tilde{\Delta} = \tilde{\Delta}_1$ and if the Clifford algebras $C(Q)$ and $C(Q_1)$ are isomorphic, $Q$ and $Q_1$ are equivalent. (Consider first the case where $-\Delta$ is a square in $K$, and show that in this case $C^+(Q)$ and $C^+(Q_1)$ are isomorphic; then reason as in Exercise 14 a), using a) and Exercise 6 of chap. VIII, § 11. In the general case, use Exercise 12 $a))$.

(c) Now prove that the special Clifford group $G^+$ (for the form $Q$) is identical with the group of invertible elements of $C^+(Q)$. (If $(e_1, e_2, e_3)$ is an orthogonal basis of $E$, and if $j = e_1 e_2 e_3$ in $C(Q)$, observe that $x \to xj$ is an isomorphism of vector spaces of $E$ onto $P$).

(d) Deduce from a) and c) that if $Q$ has index 1, the group of rotations $O^+(Q)$ is isomorphic to the projective group $\mathbf{PGL}_2(A)$ (chap. II, 2e éd., App. III, No. 6).

¶ 16) We retain the general hypotheses and notation of Exercise 14, but suppose $n = 4$.

a) Give an example where $\tilde{\Delta} = \tilde{\Delta}_1$ and where $C(Q)$ and $C(Q_1)$ are isomorphic, but where $Q$ and $Q_1$ are not equivalent (cf. exerc. 7).

b) Let $(e_i)_{1 \leq i \leq 4}$ be an orthogonal basis of $E$ for $Q$, $Q_0$ the restriction of $Q$ to the hyperplane $H = Ae_1 + Ae_2 + Ae_3$. Now prove that, if $Z$ is the center of $C^+(Q)$, the algebra $C^+(Q)$ is isomorphic to the tensor product $Z \otimes_A C^+(Q_0)$. For all $z \in C^+(Q)$, one has $\beta(z)z \in Z$; for $z$ to belong to the special Clifford group $G^+$, it is necessary and sufficient that $z$ be invertible and that $\beta(z)z \in A.1$. Deduce that the group $O_0^+(Q)$ is isomorphic to the quotient by $\{1, -1\}$ of the group of elements $z \in Z \otimes_A C^+(Q_0)$ such that $\beta(z)z = 1$.

c) Assume that $\Delta$ is not a square in $A$ (which implies, by virtue of Witt's theorem, that the index of $Q$ is 0 or 1). If $Q_0'$ is the quadratic form obtained from $Q_0$ by extension to $A' = A(\sqrt{\Delta})$ of the field of scalars, deduce from b) that $O_0^+(Q)$ is isomorphic to $O_0^+(Q_0')$. In particular, if $Q$ has index 1, $O_0^+(Q)$ is the group of commutators of $O(Q)$ and is isomorphic to $\mathbf{PSL}_2(A')$ (cf. exerc. 15 $d$) and chap. III, § 7, exerc. 8).

d) We suppose that $\Delta$ is a square in $A$ (which implies, by virtue of Witt’s theorem, that $Q$ is of index 0 or 2) and that $Q(e_4) = 1$. If we put $j = e_1 e_2 e_3$, every $x \in E$ can be written in one and only one way $x = \alpha e_4 + jz$, where $\alpha \in A$ and $z$ is a pure quaternion (Exercise 15 $a$)) in $L = C^+(Q_0)$; if we put $\psi(x) = \alpha + z$, $\psi$ is an isomorphism of vector spaces from $E$ onto $L$. Let $Z = Ac' + Ac''$, where $c'$ and $c''$ are the two orthogonal idempotents in $Z$; every invertible element $s \in C^+(Q)$ can be written in one and only one way $s = uc' + vc''$, where $u$ and $v$ belong to $L$; for $s$ to belong to the special Clifford group $G^+$, it is necessary and sufficient that $u \bar{u} = v \bar{v}$, and we then have $\psi(sxs^{-1}) = u \psi(x) v^{-1}$ for every $x \in E$. Deduce that the quotient of $O_0^+(Q)$ by its center (which is a group with 2 elements, cf. Exercise 11 $c$)) is isomorphic to the product $O_0^+(Q_0) \times O_0^+(Q_0)$; in particular, if $Q$ is of index 2, this quotient group is isomorphic to $\mathbf{PSL}_2(A) \times \mathbf{PSL}_2(A)$.

17) Let K be a commutative field of characteristic $\neq 2$, $A$ the field $K(X_n)_{n \in \mathbf{N}}$ of rational fractions over K, with respect to a countable family of indeterminates (Chapter IV, § 3, No. 1). Let E be a vector space over A, having a countable basis $(e_n)_{n \in \mathbf{N}}$, and let $\Phi$ be a symmetric bilinear form on E, for which $(e_n)$ is an orthogonal basis and such that $\Phi(e_n, e_n) = X_n$ for every $n \in \mathbf{N}$. If we put $Q(x) = \Phi(x, x)$, show that the Clifford algebra C(Q) is a field (cf. Chapter VIII, § 12, Exercise 14).