Exercise 11, § 6

(a) Let g be a Lie algebra, r its radical, s a Levi subalgebra of g and m an ideal of g containing r. There exists an ideal t of s, supplementary to m in g, such that [m, t] ⊂ r. (b) Let g be a Lie algebra and a a subinvariant subalgebra of g. Then there exists a composition series g = g_0 ⊃ g_1 ⊃ ... ⊃ g_k = a such that g_i is the direct sum of g_{i+1} and a subalgebra h_i which is either 1-dimensional and contained in the radical r(g_i) of g_i or simple and such that [h_i, g_{i+1}] ⊂ r(g_{i+1}). (Reduce it to the case where a is an ideal of g. Let g' = g/a and r' be the radical of g'. The algebra g'/r' is the product of its simple ideals a_1, a_2, ..., a_c.

Take a composition series of $g'$ consisting of a composition series of $r'$ whose successive quotients are 1-dimensional and the inverse images of the ideals $a_1, a_1 \times a_2, \ldots, a_1 \times a_2 \times \cdots \times a_c$. Then take the inverse image in $g$ of this composition series.)