Corollary

Let $\gamma$ be a linear combination of roots with integer coefficients, and $\alpha$ an indivisible root. If $\gamma$ is proportional to $\alpha$, then $\gamma \in \mathbf{Z}\alpha$.

By Prop. 15 of no. 5, $C$ can be chosen so that $\alpha \in B(C)$. By Th. 3,

Thus, if $\gamma$ is proportional to $\alpha$, then $\gamma = n_{\alpha} \alpha$, which proves the corollary.

Now let $S$ be the set of reflections $s_{\alpha}$ for $\alpha \in B(C)$ and let $T$ be the union of the conjugates of $S$ under $W$. For $\alpha \in B(C)$ and $w \in W$, the element $t = ws_{\alpha}w^{-1}$ of $T$ is the orthogonal reflection $s_{\beta}$ associated to the root $\beta = w(\alpha)$; conversely, for any indivisible root $\beta$, there exists an element $w \in W$ such that $\alpha = w^{-1}(\beta) \in B(C)$ (Prop. 15) and $s_{\beta} = ws_{\alpha}w^{-1} \in T$. It follows that a bijection $\psi$ from the set of indivisible roots to $\{\pm 1\} \times T$ is obtained by associating to an indivisible root $\beta$ the pair $(\varepsilon, s_{\beta})$, where $\varepsilon = +1$ if $\beta$ is positive and $\varepsilon = -1$ if $\beta$ is negative.

On the other hand, $(W, S)$ is a Coxeter system (Th. 2) and the results of Chap. IV, § 1, no. 4 can be applied. We have seen that, if $w$ is an element of $W$ of length (with respect to $S$) equal to $q$, there exists a subset $T_w$ of $T$, with $q$ elements, such that, if $w = s_1 \ldots s_q$ with $s_i \in S$ and if

(for $1 \leq i \leq q$), then $T_w = \{t_1, \ldots, t_q\}$. Recall that we have also defined in no. 4 of § 1 a number $\eta(w, t)$ (for $w \in W$ and $t \in T$) equal to $+1$ if $t \notin T_w$ and to $-1$ if $t \in T_w$. Finally, recall that, if we define a map $U_w$ from the set $\{\pm 1\} \times T$ to itself by the formula the map $w \mapsto U_w$ is a homomorphism from $W$ to the group of permutations of the set $\{\pm 1\} \times T$ (Chap. IV, § 1, no. 4, Lemma 1).