Corollary 1

*Suppose that K is of characteristic $\neq 2$. Let E be a finite-dimensional vector space over K and $\phi$ a non-degenerate symmetric (resp. alternating) bilinear form on E. For all $u \in \mathscr{L}(E)$, let $u^*$ be the adjoint of u relative to $\phi$. Let G be the orthogonal (resp. symplectic) group of $\phi$. Then G is a Lie subgroup of $\mathbf{GL}(E)$ and $L(G)$ is the set of $x \in \mathscr{L}(E)$ such that $x^* = -x$.

We apply Proposition 37 with $A = \mathscr{L}(E)$ and $x^t = x^*$.*