Proposition 3

There exists a unique homomorphism

such that $\sigma(g_s) = \sigma_s$ for all $s \in S$. The elements of $\sigma(W)$ preserve the bilinear form $B_M$.

To prove the existence and uniqueness of $\sigma$, it is enough to show that $(\sigma_s \sigma_{s'})^{m(s, s')} = 1$ if $m(s, s') \neq +\infty$. Now, if $s = s'$, this follows from the fact that $\sigma_s$ is of order 2; if $s \neq s'$, it follows from what we proved in no. 2. Finally, since the reflections $\sigma_s$ preserve $B_M$, so do the elements of $\sigma(W)$.