Remark 27

Let K be a maximal ordered field, E a vector space of dimension n over K, $\Phi$ a non-degenerate symmetric bilinear form on E, of signature (s, t) distinct from (n, 0) and from (0, n). Let $(e_i)$ be an orthogonal basis for $\Phi$, such that $\Phi(e_i, e_i) = 1$ for $1 \leq i \leq s$, $\Phi(e_i, e_i) = -1$ for $s+1 \leq i \leq n$. For every orthogonal transform $u \in \mathbf{U}(\Phi)$, let

be the matrix of u with respect to $(e_i)$, written in the form of a square tableau of matrices corresponding to the partition of $(1, n)$ into $(1, s)$ and $(s+1, n)$.

a) Prove the relations

b) Let R be a matrix over K with t rows and s columns, such that $I_s - {}^tR \cdot R$ is the matrix of a positive non-degenerate hermitian form (on $K^s$). Show that $\det(M + \lambda NR)$ does not change sign for $-1 \leq \lambda \leq 1$ in K (show, using a), that ${}^t(M + \lambda NR)(M + \lambda NR)$ is the matrix of a positive non-degenerate symmetric form).

c) Put $\sigma(u) = \sigma(U) = \operatorname{sgn}(\det M)$; show that, for any two elements $u, v$ of the orthogonal group $\mathbf{U}(\Phi)$, one has $\sigma(uv) = \sigma(u)\sigma(v)$ (use a) and b)). Deduce that the commutator group of the special orthogonal group $\mathbf{SU}(\Phi)$ is distinct from $\mathbf{SU}(\Phi)$ (cf. § 10, exerc. 9).