Proposition 4

Let u ∈ GL(E) be an involution. The following properties are equivalent: a) u belongs to the unitary group associated with Φ ; b) the submodules U^+ = $\frac{1}{2}(1 + u)(E)$ and U^- = $\frac{1}{2}(1 - u)(E)$ are orthogonal (and consequently non-isotropic). Moreover, if A is a field and E is finite-dimensional, properties a) and b) are equivalent to: c) u = u*. Indeed, for x ∈ U^+ and y ∈ U^-, the relation $\Phi(u(x), u(y)) = \Phi(x, y)$ gives $2\Phi(x, y) = 0$, hence a) implies b). Conversely, we obviously have $\Phi(u(x), u(y)) = \Phi(x, y)$ when x and y both belong to U^+ or both belong to U^-, and, in view of b), this relation is still true when one of them belongs to U^+ and the other to U^-; since E is the direct sum of U^+ and U^-, we see that b) implies a). Finally, when E is a finite-dimensional vector space, the adjoint u* is defined since Φ is non-degenerate; relation a) is equivalent to $uu^* = 1$ (§ 1, No. 8, cor. of prop. 8); since $u^2 = 1$ by assumption, a) and c) are equivalent.