We denote by $G_0$ the subgroup of $G$ formed by elements $g$ such that $g(\mathcal{H})=\mathcal{H}$. For all $w\in V$ such that $B(w,w)>0$, we define the linear map
$$s_w : v \mapsto v - 2\frac{B(v,w)}{B(w,w)}w.$$
Show that $s_w^2 = \mathrm{Id}_V$, and determine the eigenvalues and eigenspaces of $s_w$.