Exercise 16, § 6

Let $g$ be a subalgebra of $\mathfrak{gl}(n, K)$ such that the identity representation of $g$ is semi-simple. Show that every nilpotent matrix $X \in g$ is contained in a 3-dimensional simple Lie subalgebra of $g$. (Show that $g$ is reductive in $\mathfrak{gl}(n, K)$; then use Exercises 14 and 15 (c).)