Corollary

Let $\sigma$ be half the sum of the elements $> 0$ of $R^-$ (for $B^-$). For all $\alpha \in V$, the sum of the coordinates of $\alpha$ with respect to the basis $B$ is $\langle \alpha, \sigma \rangle$. If $\alpha \in R$, this sum is equal to $\frac{1}{2} \sum_{\beta \in R_+ (\mathbf{C})} n(\alpha, \beta)$.

Interchanging the roles of $R$ and $R^-$ above, we have $\langle \alpha, \sigma \rangle = 1$ for all $\alpha \in B$, hence the corollary.