Exercise 1, § 2

Let $\mathfrak{g}$ be a simple Lie algebra of dimension 3, and $\Phi$ its Killing form (cf. §1, Exerc. 15, 16, 17).

a) An element $x\in \mathfrak{g}$ is regular if and only if $\Phi (x, x)\not= 0$. Let $\mathfrak{h}_x=kx$ be the Cartan subalgebra generated by such an element. Show that $\mathfrak{h}_x$ is splitting if and only if $2\Phi (x, x)$ is a square in $k$.

b) Show that

$\mathfrak{g}$ is splittable $\Leftarrow \Rightarrow \mathfrak{g}$ is isomorphic to $\mathfrak{s}\mathfrak{l}(2, k)$.