Contragredient lie algebra before the maximal ideal quotient

Fix a minimal realization of $A$ using Minimal realizations exist and are unique up to isomorphism. The universal contragredient algebra $\tilde{\mathfrak{g}}(A)$ is the free Lie algebra on a basis of $\mathfrak{h}$ and symbols $e_i,f_i$, modulo $[h,h^{\mathrm{\prime }}]=0$, $[h,e_i]={\alpha }_i(h)e_i$, $[h,f_i]=-{\alpha }_i(h)f_i$ and $[e_i,f_j]={\delta }_{ij}{h}_i$, for all $h,h^{\mathrm{\prime }}\in \mathfrak{h}$.

The free algebra is constructed in Free Lie construction for finite Kac Moody generators. Any linear map on $\mathfrak{h}$ and images of the symbols satisfying these relations extend uniquely to a Lie homomorphism from this quotient. Give $h,e_i,f_i$ degrees $0,{\alpha }_i,-{\alpha }_i$ in Kac Moody root lattice height and positive cone; all relations are homogeneous. Write ${\tilde{\mathfrak{n}}}^{+}$ and ${\tilde{\mathfrak{n}}}^{-}$ for the subalgebras generated by the $e_i$ and $f_i$. Their freeness and injectivity of the Cartan map are justified in Contragredient algebra has a triangular decomposition ↗, not imposed as additional relations.

Source comparison: Kleshchev, Definition 1.3.1, p.13.