Proposition 29

Let $\rho$ be half the sum of the roots $> 0$.

(i) $\rho = \sum_{\alpha \in B} \overline{\omega}_\alpha$; this is an element of $C$. (ii) $s_\alpha(\rho) = \rho - \alpha$ for all $\alpha \in B$. (iii) $(2\rho|\alpha) = (\alpha|\alpha)$ for all $\alpha \in B$.

Since $R$ is reduced, $s_\alpha(R_+(C) - \{\alpha\}) = R_+(C) - \{\alpha\}$ and $s_\alpha(\alpha) = -\alpha$ for $\alpha \in B$ (no. 6, Cor. 1 of Prop. 17), so $s_\alpha(2\rho) = 2\rho - 2\alpha$. Since $s_\alpha(\rho) = \rho - \langle \rho, \alpha^\vee \rangle$, we see that

Hence, $\rho = \sum_{\beta} \overline{\omega}_\beta$, and consequently $\rho \in C$. Finally, (iii) is equivalent to $\langle \rho, \alpha^\vee \rangle = 1$.