Proposition 1

The group $W_a$ is the semi-direct product of W by Q.

Since W is generated by the reflections $s_\alpha$, it is contained in $W_a$. On the other hand, $t(\alpha^\sim) = s_{\alpha,1} \circ s_\alpha$ if $\alpha \in \mathbf{R}$, which shows that $Q \subseteq W_a$.

Since W leaves $Q(\mathbf{R}^\sim)$ stable (§ 1, no. 9), the group G of affine transformations generated by W and Q is the semi-direct product of W by Q. Now $G \subseteq W_a$ from above and $s_{\alpha,k} \in G$ for all $\alpha \in \mathbf{R}$ and $k \in \mathbf{Z}$ by (1). It follows that $W_a = G$.