Theorem 3

Every root is a linear combination with integer coefficients of the same sign of elements of $B(C)$. In particular, every root is either positive or negative for $C$.

If $\alpha \in R$, the kernel $L_\alpha$ of $\alpha$ does not meet $C^*$, so $\alpha$ is either $> 0$ on the whole of $C^*$ or $< 0$ on the whole of $C^*$, hence the second assertion. It remains to show that $\alpha$ is contained in the subgroup $P$ of $V$ generated by $B(C)$; we can assume that $\alpha$ is indivisible. Now the group $P$ is clearly stable under the $s_{\gamma}$, for $\gamma \in B(C)$, hence also under $W(R)$ by Th. 2. Since $\alpha$ is of the form $w(\beta)$, with $w \in W(R)$ and $\beta \in B(C)$ (cf. Prop. 15), we have $\alpha \in P$. Q.E.D.

Denote by $R_+(C)$ the set of roots that are positive for $C$. Thus,

is a partition of $R$.