Proposition 6

Let g be a Lie algebra, r its radical and n its largest nilpotent ideal. Every derivation of g maps r into n.

Let D be a derivation of g. Let g' = g + Kx_0 be a Lie algebra in which g is an ideal of codimension 1 such that Dx = [x_0, x] for all x ∈ g (§ 1, no. 8, Example 1). By Corollary 3 to Proposition 5, r is contained in the radical r' of g'. Then D(r) = [x_0, r] ⊂ [g', g'] ∩ r' = s'. For all x ∈ s', ad_{g'} x is nilpotent (Theorem 1). Hence, for all x ∈ s' ∩ g, ad_g x is nilpotent. Hence D(r) is contained in the nilpotent ideal s' ∩ g of g.