Exercise 7, § 6

(a) Let g be a Lie algebra, h a semi-simple ideal of g and a the centralizer of h in g, so that g is identified with h × a. Show that, for every ideal t of g, then t = (t ∩ h) × (t ∩ a). (Let t_1 be the canonical projection of t onto h; it is an ideal of h and hence Dt_1 = t_1; deduce that t_1 ⊂ t.) (b) Let β be an invariant bilinear form on g. Show that h and a are orthogonal with respect to β (use the fact that h = Dh and that β = β_1 + β_2, where β_1 (resp. β_2) is an invariant bilinear form whose restriction to a (resp. h) is zero. (c) Deduce from (a) that there exists in g a largest semi-simple ideal. (Consider a maximal semi-simple ideal of g.)