Let $G$ be the group of endomorphisms $g$ of $V$ such that $B(gu,gv)=B(u,v)$ for all $u,v\in V$. Show that for all $g\in G$, we have $g(\mathcal{H})=\mathcal{H}$ or $-g(\mathcal{H})=\mathcal{H}$.
Let $G$ be the group of endomorphisms $g$ of $V$ such that $B(gu,gv)=B(u,v)$ for all $u,v\in V$.
Show that for all $g\in G$, we have $g(\mathcal{H})=\mathcal{H}$ or $-g(\mathcal{H})=\mathcal{H}$.