Serre Lie algebra of a finite-type Cartan matrix

Let $A=(a_{ij})$ be a finite-type Cartan matrix of size $r$ (Properties of finite-type Cartan matrices) and let $V$ be the complex vector space with basis $e_1,\dots ,e_r,f_1,\dots ,f_r,h_1,\dots ,h_r$. The Serre Lie algebra $\mathfrak{g}(A)$ is the Lie algebra presented by the generators $e_i,f_i,h_i$ and the relations $[h_i,h_j]=0,[h_i,e_j]=a_{ij}{e}_j,[h_i,f_j]=-a_{ij}{f}_j,[e_i,f_j]={\delta }_{ij}{h}_i,$ together with the Serre relations $(\mathrm{ad}{e}_i{)}^{1-a_{ij}}{e}_j=0,(\mathrm{ad}{f}_i{)}^{1-a_{ij}}{f}_j=0(i\mathrm{\ne }j),$ in the sense of Lie algebra presented by generators and relations. The exponents are positive integers because $a_{ij}\le 0$, and for $a_{ij}=0$ the Serre relations reduce to $[e_i,e_j]=0$ and $[f_i,f_j]=0$. The relations express the standard presentation of a complex semisimple Lie algebra relative to simple-root ${\mathfrak{s}\mathfrak{l}}_2$ triples; the algebra $\mathfrak{g}(A)$ is shown to be finite-dimensional semisimple with Cartan matrix $A$ in Serre presentation theorem.