Ideals and quotients of semisimple Lie algebras

Let $p_i$ be projection to ${\mathfrak{g}}_i$. If $p_i(\mathfrak{a})\mathrm{\ne }0$, simplicity and nonabelianness give $[{\mathfrak{g}}_i,p_i(\mathfrak{a})]={\mathfrak{g}}_i$. But bracketing an element supported in the $i$th factor with an element of $\mathfrak{a}$ produces an element of $\mathfrak{a}$ supported only in that factor. Hence ${\mathfrak{g}}_i\subseteq \mathfrak{a}$. If the projection is zero, the factor does not occur. Therefore $\mathfrak{a}$ is exactly the direct sum of the factors for which its projection is nonzero.

The sum of the remaining factors is an ideal complement $\mathfrak{b}$, so $\mathfrak{g}=\mathfrak{a}\oplus \mathfrak{b}$. Both $\mathfrak{a}$ and $\mathfrak{b}$ are direct sums of simple ideals and hence semisimple. By [L2], projection restricts to an isomorphism $\mathfrak{b}\cong \mathfrak{g}\mathrm{/}\mathfrak{a}$, so the quotient is semisimple as well.

For $\mathfrak{g}=0$, the indexing family and both subfamilies are empty; the ideal and quotient are zero. The cases $\mathfrak{a}=0$ and $\mathfrak{a}=\mathfrak{g}$ correspond to the empty and full subfamilies and are included in step 2.1.