Exercise 10, § 5

Let $k'$ be an extension of $k$ and $\mathfrak{g}'$ a $k'$-Lie subalgebra of

a) Show that there exists a smallest Lie subalgebra $\mathfrak{g}$ of $\mathfrak{g}\mathfrak{l}(V)$ such that $\mathfrak{g}\otimes_kk'$ contains $\mathfrak{g}'$.

b) Assume that $k'$ is algebraically closed and denote by G the group of $k$-automorphisms of $k'$; this group operates in a natural way on $V\otimes_kk'$. Show that $\mathfrak{g}\otimes_kk'$ is the Lie subalgebra generated by the conjugates of $\mathfrak{g}'$ under G (use the fact that $k$ is the field of G-invariants in $k'$).

c) Show that $\mathfrak{g}$ is decomposable if $\mathfrak{g}'$ is decomposable. (Reduce to the case where $k'$ is algebraically closed, and use $b$) as well as Cor. 1 and 3 of Th. 1.)