Exercise 11, § 2

Let R be a root system of rank $l$, and $\Phi_R$ the corresponding canonical bilinear form (Chap. VI, §1, no. 12). The matrix $\Phi = (\Phi_R(\alpha , \beta ))_{\alpha ,\beta\in R}$ is of rank $l$, and $\Phi^2=\Phi$. Deduce the formula $\sum_{\alpha\in R}\Phi_R(\alpha , \alpha ) =$ Tr$\Phi =l$.