Serre presentation of a kac moody algebra

Let $S^{+}$ be the ideal in the free positive half generated by the positive Serre vectors. The adjoint rank-one calculation gives $[f_k,s_{ij}]=0$ for each such generator already in $\tilde{\mathfrak{g}}$: its coefficient for $k=i$ is $(1-a_{ij})(-a_{ij}-(1-a_{ij})+1)=0$, and for $k=j$ the only exponent-one boundary uses $a_{ji}=0$. For other $k$ it is zero by the mixed simple relations. Cartan brackets scale each homogeneous generator. Jacobi then proves by induction on positive adjoints that $S^{+}$ is stable under all negative simple generators; hence it is an ideal of the whole universal algebra contained in its positive half. The same holds for $S^{-}$. Therefore the ideal generated in the whole algebra by both families is exactly $S^{+}\oplus S^{-}$.

F1 identifies that whole ideal with $\mathfrak{r}$. Intersect its direct sum in step 1.1 with each free half to obtain $S^{\pm }={\mathfrak{r}}^{\pm }$. Quotienting the original triangular direct sum therefore gives the stated separate half presentations and the full Cartan–Serre presentation. Order the resulting homogeneous bases negative, Cartan, positive. F2 identifies the tensor product of their three ordered monomial bases bijectively with the ordered basis of $U(\mathfrak{g})$, proving the multiplication isomorphism.