Corollary 1

A semi-simple Lie algebra is the product of its simple ideals $g_i$. Every ideal of g is the product of certain of the $g_i$.

$g = a_1 \times \cdots \times a_m$, where the $a_i$ are simple. As the centre of $a_i$ is zero, the centralizer of $a_i$ in g is the product of the $a_j$ for $j \neq i$. Then let a be an ideal of g. If it does not contain $a_i$, then $a \cap a_i = \{0\}$, hence $[a, a_i] = \{0\}$ and a is contained in the product of the $a_j$ for $j \neq i$. It follows that a is the product of certain of the $a_i$. Hence the simple ideals of g are precisely the $a_i$.

The simple ideals of a semi-simple Lie algebra are called the simple components of g.