Exercise 2, § 2

Let $k_1$ be an extension of $k$ of finite degree $n\geq 2$.

a) Show that the semi-simple $k$-algebra $\mathfrak{s}\mathfrak{l}(2, k_1)$ is not splittable.

b) Show that the splittable simple $k$-algebra $\mathfrak{s}\mathfrak{l}(n, k)$ contains a Cartan subalgebra $\mathfrak{h}_1$ that is not splitting. (Choose an embedding of the algebra $k_1$ into $\mathbf{M}_n(k)$, and take $\mathfrak{h}_1=k_1\cap \mathfrak{s}\mathfrak{l}(n, k).$)