Proposition 2

For a Lie algebra g to be semi-simple, it is necessary and sufficient that it be a product of simple algebras.

The condition is sufficient (no. 1, Remark 3). Conversely, suppose that g is semi-simple. Since the adjoint representation of g is semi-simple, g is the direct sum of minimal non-zero ideals $a_1, \ldots, a_m$. Then g is identified with the product algebra of the $a_i$ (§ 1, no. 1). Every ideal of $a_i$ is then an ideal of g and hence zero or equal to $a_i$. On the other hand $a_i$ is non-commutative. Hence the $a_i$ are simple Lie algebras.