Exercise 4, § 4

Let $\mathfrak{g}$ be a splittable semi-simple Lie algebra, and $\sigma$ an automorphism of $k$. Let $\mathfrak{g}_{\sigma}$ be the Lie algebra obtained from $\mathfrak{g}$ by extending scalars by means of $\sigma$. Show that $\mathfrak{g}_{\sigma}$ is isomorphic to $\mathfrak{g}$. (Use the Cor. of Prop. 4.)