Lemma 10

Let $\mu\in Q$, and assume that $\mu$ is not a multiple of a root. There exists $w\in W(R)$ such that certain of the coordinates of $w\mu$ with respect to the basis B are $>0$ and certain of them are $<0$.

Let $V_{\mathbf{R}}$ be the vector space $Q\otimes_{\mathbf{Z}}\mathbf{R}$, in which R is a root system. By the assumption, there exists $f\in V^*_{\mathbf{R}}$ such that $\langle f, \alpha \rangle \not= 0$ for all $\alpha \in R$, and $\langle f, \mu\rangle = 0$. There exists a chamber C of $R^{\vee}$ such that $f\in C$. By Chap. VI, §1, no. 5, Th. 2 (i), there exists $w\in W(R)$ such that $wf$ belongs to the chamber associated to B, in other words such that $\langle wf, \alpha \rangle >0$ for all $\alpha \in B$. Write $w\mu=\sum_{\alpha\in B}t_{\alpha}\alpha$. Then

which proves that certain $t_{\alpha}$ are $>0$ and others are $<0$.