Proposition 2

a) The product of a finite number of Lie algebras is a compact Lie algebra if and only if each factor is compact.

b) A subalgebra of a compact Lie algebra is compact.

c) Let $\mathfrak{h}$ be an ideal of a compact Lie algebra $\mathfrak{g}$. Then the algebra $\mathfrak{g}/\mathfrak{h}$ is compact and the extension $\mathfrak{h}\rightarrow \mathfrak{g}\rightarrow \mathfrak{g}/\mathfrak{h}$ is trivial.

Assertions a) and b) follow from the characterization (iii) of Prop. 1. Part c) follows from a) and the fact that, in a reductive Lie algebra, every ideal is a direct factor (Chap. I, §6, no. 4, Cor. of Prop. 5).