Corollary

*Let D and D' be two lines of V and let $\theta$ be a measure of the angle $(\widehat{D}, \widehat{D'})$. Then $s_{D'} s_D = \rho_{2\theta}$.

We know that $s_{D'} s_D$ is a rotation since it is of determinant 1. Let $\Delta$ and $\Delta'$ be two half-lines starting at the origin carried by D and D'. We have

hence the corollary.

Now let $\Delta$ and $\Delta'$ be two half-lines of V such that

and let s and $s'$ be the orthogonal reflections with respect to the lines D and D' containing $\Delta$ and $\Delta'$. Let $\theta$ be the principal measure of the angle $(\widehat{D}, \widehat{D'})$. If $\theta \in \pi \mathbf{Q}$, denote by m the smallest integer $\geq 1$ such that $m\theta \in \pi \mathbf{Z}$. If $\theta \notin \pi \mathbf{Q}$, put $m = \infty$. Let W be the group generated by s and $s'$.