Exercise 3, § 2

Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra, R its root system, and K the restriction of the Killing form of $\mathfrak{g}$ to $\mathfrak{h}$. With the notations of Chap. VI, §1, no. 12, we have $K = B_{R^{\vee}}$ and $K = 4\gamma (R)\Phi_{R^{\vee}}$ if R is irreducible; moreover, if all the roots of R have the same length, then

$K(H_{\alpha}, H_{\alpha}) = 4h$ for all $\alpha \in R$,

where $h$ is the Coxeter number of W(R).

If for example $\mathfrak{g}$ is simple of type $E_6,E_7$ or $E_8, K(H_{\alpha}, H_{\alpha})$ is equal to $48,72$ or 120.