Finite type kac moody algebras recover the dg semisimple algebras

Let $I_1,\dots ,I_s$ be the connected components of the nonzero-entry graph, giving diagonal blocks $A_k$. By F2 each is symmetrizable and invertible, so $A$ is too and its Cartan is precisely the span of the $h_i$. F3 proves finite dimension for $A$ and for each block. F4 makes each $\mathfrak{g}(A_k)$ nonabelian simple.

For indices in different components, the positive and negative Serre exponents are one, so F1 gives $[e_i,e_j]=[f_i,f_j]=0$. The mixed relations give $[e_i,f_j]=0$, and the Cartan pairings give zero brackets of each component Cartan with generators of another. The Cartans commute. Jacobi then makes the whole subalgebras generated by different components commute. Map every generator of $\mathfrak{g}(A)$ to its component generator in ${⨁}_k\mathfrak{g}(A_k)$; all relations of F1 hold there. Conversely the component generator maps satisfy their own presentations and their images commute, giving a homomorphism from the direct sum back to $\mathfrak{g}(A)$. Both composites fix every generator, so both are identities.

The direct sum in step 2.1 consists of the simple algebras from step 1.1, proving the stated meaning of semisimple. Its root decomposition is the original one, with positive roots in the nonnegative span of the independent simple roots and with simple root spaces nonzero; F3 identifies all roots by Weyl descent. The relation ${\alpha }_j(h_i)=a_{ij}$ fixes the row/coroot convention. Finally F1 says precisely that every assignment of these generators in a complex Lie algebra satisfying these relations extends uniquely to a homomorphism; this is the intrinsic universal presentation claimed.