Theorem 2

(i) The group $W(R)$ acts simply-transitively on the set of chambers.

(ii) Let $C$ be a chamber. Then $\overline{C}$ is a fundamental domain for $W(R)$.

(iii) $C$ is an open simplicial cone (Chap. V, § 1, no. 6).

(iv) Let $L_1, L_2, \ldots, L_l$ be the walls of $C$. For all $i$, there exists a unique indivisible root $\alpha_i$ such that $L_i = L_{\alpha_i}$ and such that $\alpha_i$ is on the same side of $L_i$ as $C$.

(v) The set $B(C) = \{ \alpha_1, \ldots, \alpha_l \}$ is a basis of $V$.

(vi) $C$ is the set of $x \in V$ such that $\langle \alpha_i, x \rangle > 0$ for all $i$ (or, equivalently, the set of $x \in V$ such that $(x|\alpha_i) > 0$ for all $i$).

(vii) Let $S$ be the set of the $s_{\alpha_i}$. The pair $(W(R), S)$ is a Coxeter system (Chap. IV, § 1, no. 3).

Assertions (i) and (vii) follow from Chap. V, § 3, no. 2, Th. 1. Assertion (ii) follows from Chap. V, § 3, no. 3, Th. 2. Assertion (iv) is clear. The root $\alpha_i$ is orthogonal to $L_i$, and $\alpha_i^-$ is identified with $2\alpha_i / (\alpha_i|\alpha_i)$. Since $W(R)$ is essential (no. 1, Remark 3), assertions (iii), (v) and (vi) follow from Chap. V, § 3, no. 9, Prop. 7.