Exercise 15, § 1

Let $\mathfrak{g}$ be a Lie algebra of dimension 3. Show the equivalence of the following conditions (cf. Chap. I, §6, Exerc. 23).

(i) $\mathfrak{g}= [\mathfrak{g},\mathfrak{g}]$.

(ii) The Killing form of $\mathfrak{g}$ is non-degenerate.

(iii) $\mathfrak{g}$ is semi-simple.

(iv) $\mathfrak{g}$ is simple.