Proposition 4

Assume that V is finite dimensional. Let B be a non-degenerate symmetric bilinear form on V, X a subspace of V and $X^0$ the orthogonal complement of X with respect to B; finally, let s be the orthogonal reflection with respect to a non-isotropic hyperplane H of V. The following conditions are equivalent:

(i) X is stable under s. (ii) $X^0$ is stable under s. (iii) H contains X or $X^0$.

We have $V_s^+ = H$, and by what we have said, $V_s^-$ is the orthogonal complement $H^0$ of H with respect to B. By Prop. 3, X is stable under s if and only if $X \subset H$ or $H^0 \subset X$; but the relation $H^0 \subset X$ is equivalent to $X^0 \subset H$ by Algebra, Chap. IX, § 1, no. 6, Cor. 1 to Prop. 4. This proves the equivalence of (i) and (iii); that of (ii) and (iii) follows by interchanging the roles of X and $X^0$, since $(X^0)^0 = X$.