Exercise 4, § 6

Let $g$ be a semi-simple Lie algebra and $\rho$ a non-zero simple representation of $g$ on a finite-dimensional space $M$. Let $h$ be the corresponding semi-direct product. Show that $h = \mathscr{D}h$, that the centre of $h$ is zero and that $h$ is not the product of a semi-simple algebra and a solvable algebra.