Proposition 16

Let $B$ be a basis of $R$, and $G$ the subgroup of $A(R)$ consisting of the elements leaving $B$ stable. Then $W(R)$ is a normal subgroup of $A(R)$ and $A(R)$ is the semi-direct product of $G$ and $W(R)$.

If $\alpha \in R$ and $t \in A(R)$, then $ts_\alpha t^{-1} = s_{t(\alpha)}$; since $W(R)$ is generated by the $s_\alpha$, we see that $W(R)$ is a normal subgroup of $A(R)$. By transport of structure, $A(R)$ transforms a basis of $R$ to a basis of $R$. Since $W(R)$ acts simply-transitively on the set of bases, every element of $A(R)$ can be written uniquely in the form $g_1 g_2$, where $g_1 \in W(R)$ and $g_2 \in G$.