Exercise 2, § 6

Let $g$ be a Lie algebra, $r$ its radical and $(a_0, a_1, \ldots)$ a sequence of ideals of $g$ defined as follows: (1) $a_0 = \{0\}$; (2) $a_{i+1}/a_i$ is a maximal commutative ideal of $g/a_i$. Let $p$ be the smallest integer such that $a_p = a_{p+1} = \cdots$. Show that $r = a_p$. (Show that $g/a_p$ is semi-simple.)