We consider the three vectors
$$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$
Let $S_{1,2}$ be the subgroup of $\Gamma$ generated by $s_{w_1}$ and $s_{w_2}$. Let $v\in\mathcal{H}$.
Show that there exists $g\in S_{1,2}$ such that
$$B(gv,w_1)\geq 0 \quad \text{and} \quad B(gv,w_2)\geq 0.$$