Lemma 2

Assume that W is irreducible and that $\dim V \geq 2$. With the preceding notation, there exist two linearly independent vectors $z', z''$ such that (i) the plane P generated by $z', z''$ is stable under s' and s''; (ii) s'|P and s''|P are orthogonal reflections with respect to $Rz'$ and $Rz''$; (iii) $z', z'' \in \overline{C}$, and $P \cap C$ is the set of linear combinations of $z', z''$ with coefficients > 0.

Let $(e^1, \ldots, e^l)$ be the basis of V such that $(e^i|e_j) = \delta_{ij}$. Then C is the open simplicial cone determined by the $e^i$ (§ 3, no. 9, Prop. 7). It is clear that $V'$ is generated by $e^{r+1}, \ldots, e^l$ and $V''$ by $e^1, \ldots, e^r$. Let q be the endomorphism of V such that $q(e^1) = e_1, \ldots, q(e^l) = e_l$. Its matrix with respect to $(e^1, \ldots, e^l)$ is $Q = ((e_i|e_j))$. We have $(e_i|e_j) \leq 0$ for $i \neq j$ (§ 3, no. 4, Prop. 3). Since W is irreducible, there does not exist any partition $\{1, 2, \ldots, l\} = I_1 \cup I_2$ such that $(e_i|e_j) = 0$ for $i \in I_1$ and $j \in I_2$. Thus (§ 3, no. 5, Lemma 4) Q has an eigenvector $(a_1, \ldots, a_l)$ all of whose coordinates are > 0; let a be the corresponding eigenvalue. Put

and let $P$ be the plane generated by $z'$ and $z''$. Then $P \cap C$ is the set of linear combinations of $z'$ and $z''$ with coefficients $> 0$. The relation $q(z) = az$ gives $\sum_{j=1}^l a_j e_j = \sum_{j=1}^l aa_j e^j$; scalar multiplying by $e_k$ (where $k \leq r$) gives

Thus, $(a-1)z'' + z'$ is orthogonal to $e^1, \ldots, e^r$, that is, to $V''$. Hence, $s''$ leaves stable the plane generated by $z''$ and $(a-1)z'' + z'$, that is, $P$. Similarly, $s'$ leaves $P$ stable. Since $z' \in P \cap V'$ and $z'' \in P \cap V''$, $s'|P$ and $s''|P$ are the reflections with respect to $Rz'$ and $Rz''$.