Existence theorem for complex semisimple Lie algebras

If $\mathrm{\Phi }=\mathrm{\varnothing }$, then its ambient space is zero because $\mathrm{\Phi }$ spans it. Taking $\mathfrak{g}=0$ gives the empty root system and a semisimple algebra by [L5], proving the empty case. Henceforth suppose $\mathrm{\Phi }\mathrm{\ne }\mathrm{\varnothing }$.

Choose a regular vector and the resulting base $\mathrm{\Delta }$ by [L2]. By [L1], write $\mathrm{\Phi }={\mathrm{\Phi }}_1\bigsqcup \cdots \bigsqcup {\mathrm{\Phi }}_m$. The restriction of the regular vector to $E_j=\mathrm{span}{\mathrm{\Phi }}_j$ is regular for ${\mathrm{\Phi }}_j$, and positivity is tested componentwise, so $\mathrm{\Delta }$ is the disjoint union of the bases ${\mathrm{\Delta }}_j=\mathrm{\Delta }\cap {\mathrm{\Phi }}_j$. Let $A_j$ be the Cartan matrix of $({\mathrm{\Phi }}_j,{\mathrm{\Delta }}_j)$; the Cartan matrix of $\mathrm{\Phi }$ is the block diagonal matrix $\mathrm{diag}(A_1,\dots ,A_m)$.

For each $j$, [L3] gives a finite-dimensional semisimple Serre algebra $\mathfrak{g}(A_j)$ with based root system ${\mathrm{\Psi }}_j$ having Cartan matrix $A_j$. By [L4], the base-matching map is a root-system isomorphism ${\phi }_j:{\mathrm{\Phi }}_j\to {\mathrm{\Psi }}_j$. Since $A_j$ is the Cartan matrix of the irreducible component ${\mathrm{\Phi }}_j$, [L3] also makes $\mathfrak{g}(A_j)$ simple.

Put $\mathfrak{g}={⨁}_{j=1}^m\mathfrak{g}(A_j)$ and take the direct sum of the Cartan subalgebras supplied by [L3]. Brackets between distinct summands vanish, so the roots of $\mathfrak{g}$ are exactly the roots of the summands, extended by zero on the other Cartan summands; hence its root system is the orthogonal disjoint union ${\mathrm{\Psi }}_1\bigsqcup \cdots \bigsqcup {\mathrm{\Psi }}_m$. The disjoint union of the maps ${\phi }_j$ from step 1.3 is therefore an isomorphism from $\mathrm{\Phi }$ onto this root system. The direct sum is finite-dimensional and semisimple, and if $\mathrm{\Phi }$ is irreducible then $m=1$ and $\mathfrak{g}=\mathfrak{g}(A_1)$ is simple.