Exercise 5, § 4

a) Let $\mathfrak{g}$ be a simple Lie algebra, and $k_1$ the commutant of the adjoint representation of $\mathfrak{g}$. Show that $k_1$ is a commutative field, a finite extension of $k$, and that $\mathfrak{g}$ is an absolutely simple $k_1$-Lie algebra.

Conversely, if $k_1$ is a finite extension of $k$, and $\mathfrak{g}$ an absolutely simple $k_1$-Lie algebra, then $\mathfrak{g}$ is a simple $k$-Lie algebra, and the commutant of its adjoint representation can be identified with $k_1$.

b) Let $k'$ be a Galois extension of $k$ containing $k_1$. Show that $\mathfrak{g}_{(k')}$ is a product of $[k_1:k]$ absolutely simple algebras. When $\mathfrak{g}_{(k')}$ is split, these algebras are mutually isomorphic. (Use Exerc. 4.)