Corollary 1

Suppose that A is a field and that E is finite-dimensional. The mapping $u \to \frac{1}{2}(1 + u)(E)$ is a bijection from the set of involutions u belonging to the unitary group associated with Φ onto the set of non-isotropic subspaces of E; the subspace U^+ corresponding to u is the set of elements of E invariant under u.

By prop. 4, it suffices to show that every non-isotropic subspace M of E is the set of vectors invariant under an involution $u \in \mathbf{U}(\Phi)$, and that the latter is unique. Now E is the direct sum of M and M^0 (§ 4, No. 1, cor. of prop. 1), and necessarily $u(x) = x$ for $x \in M$ and $u(x) = -x$ for $x \in M^0$ by virtue of prop. 4; these relations determine u uniquely, and the endomorphism u thus determined obviously answers the question (prop. 4).

We say that the involution $u$ thus determined is the symmetry with respect to the non-isotropic subspace M.