Exercise 5, § 6

Let $a$ be a Lie algebra and $r$ its radical. If $r$ has a decreasing sequence of characteristic ideals $r = r_0 \supset r_1 \supset \cdots \supset r_n = \{0\}$ such that $\dim r_i/r_{i+1} = 1$ for $0 \leq i < n$, $g$ is the product of a semi-simple algebra and a solvable algebra.

(Let s be a Levi subalgebra of g. For all x ∈ s, let ρ(x) be the restriction of ad_gx to r. Then ρ is a direct sum of 1-dimensional representations, which are zero because s = Ds.)