Exercise 3, § 6

Assume that $\mathfrak{g}$ is simple, and denote by $\widetilde{\alpha}$ the highest root of R. The module $E( \widetilde{\alpha})$ is isomorphic to $\mathfrak{g}$, equipped with the adjoint representation. If C is the Casimir element associated to the Killing form of $\mathfrak{g}$, the image of C in End $E( \widetilde{\alpha})$ is the identity (cf. Chap. I, §3, no. 7, Prop. 12). Deduce, by using the Cor. of Prop. 7, that

where $\Phi_R$ is the canonical bilinear form on $\mathfrak{h}^*$ (Chap. VI, §1, no. 12).