Corollary 3

Let g be a Lie algebra and m an ideal of g such that g/m is semi-simple. Then g contains a subalgebra supplementary to m in g. In other words, every extension of a semi-simple Lie algebra is inessential.

Let s be a Levi subalgebra of g (Theorem 5). Its canonical image in g/m is a Levi subalgebra and therefore equal to g/m, hence g = s + m. Then an ideal of s supplementary in s to the ideal m ∩ s is a subalgebra of g supplementary to m in g.