Exercise 8, § 6

Let g be a Lie algebra. An ideal h of g is called minimal if h ≠ {0} and every ideal of g contained in h is equal to {0} or h. (a) Every simple ideal of g is minimal. (b) Let h be a minimal ideal of g and r the radical of g. Then either h ⊂ r, in which case h is abelian, or h ∩ r = {0}, in which case h is simple. (Use the fact that the derived ideal of a Lie algebra and the simple components of a semi-simple Lie algebra are characteristic ideals.)