Exercise 10, § 11

Assume that $\mathfrak{g}$ is splittable semi-simple. Let $(x, h, y)$ be an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$.

a) Show that there exists a splitting Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ containing $h$. (Use Exerc. $10b$) of §3.)

b) Choose $\mathfrak{h}$ as in a). Show that $h\in \mathfrak{h}_{\mathbf{Q}}$ and that there exists a basis B of $R(\mathfrak{g},\mathfrak{h})$ such that $\alpha (h)\in \{0,1,2\}$ for all $\alpha \in B$ (cf. Prop. 5). The element $x$ belongs to the subalgebra of $\mathfrak{g}$ generated by the $\mathfrak{g}^{\alpha},\alpha \in B$.

c) Deduce from a) and b), and the Jacobson-Morozov theorem, a new proof of the fact that every nilpotent element of $\mathfrak{g}$ is contained in a Borel subalgebra (cf. §10, Cor. 2 of Th. 1).