Theorem 1

Let g be a Lie algebra. The following conditions are equivalent:

(a) g is semi-simple. (b) The radical r of g is zero. (c) The Killing form $\beta$ of g is non-degenerate.

Moreover, a semi-simple Lie algebra is equal to its derived ideal.

(a) $\Rightarrow$ (b): for if $r \neq \{0\}$, the last non-zero derived algebra of r is a commutative ideal of g.

(b) $\Rightarrow$ (c): this follows from Proposition 5 (b) of § 5, no. 5 (which at the same time proves the last assertion of the theorem).

(c) $\Rightarrow$ (a): this follows from Proposition 6 (b) of § 4, no. 4.