Exercise 8, § 5

Let $\overline{k}$ be an algebraic closure of $k$, and $\overline{\mathfrak{g}}= \overline{k}\otimes_k\mathfrak{g}$. The Galois group Gal($\overline{k}/k$) operates naturally on $\overline{\mathfrak{g}}$, on the canonical Cartan subalgebra of $\overline{\mathfrak{g}}$ (no. 3, Remark 2), as well as on its root system $\overline{R}$ and its canonical basis $\overline{B}$. We thus obtain a continuous homomorphism (i.e. with open kernel)

$\pi :$ Gal($\overline{k}/k$)$\rightarrow$ Aut( $\overline{R},\overline{B}$).

Show that $\mathfrak{g}$ is splittable if and only if the following two conditions are satisfied:

(i) The homomorphism $\pi$ is trivial.

(ii) $\mathfrak{g}$ has a Borel subalgebra $\mathfrak{b}$.

(Show that a Cartan subalgebra $\mathfrak{h}$ contained in $\mathfrak{b}$ is splitting if and only if $\pi$ is trivial.)