Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra, let $\mathfrak h \subset \mathfrak g$ be a Cartan subalgebra, and let $\Delta \subset \mathfrak h^*$ be the corresponding root system. Fix a base of simple roots $\{\alpha_1,\dots,\alpha_n\} \subset \Delta$, and for each $i \in \{1,\dots,n\}$ choose Chevalley generators

\begin{align*} e_i \in \mathfrak g_{\alpha_i}, \qquad f_i \in \mathfrak g_{-\alpha_i}, \qquad h_i = [e_i,f_i] = \alpha_i^\vee \in \mathfrak h. \end{align*}

Let $A = (a_{ij})_{1 \le i,j \le n}$ be the Cartan matrix with

\begin{align*} a_{ij} = \langle \alpha_j,\alpha_i^\vee\rangle = \alpha_j(h_i). \end{align*}

Then, for all $i,j \in \{1,\dots,n\}$, the Chevalley generators satisfy the Serre relations

\begin{align*} [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, \end{align*}

and, for $i \ne j$,

\begin{align*} (\operatorname{ad} e_i)^{1-a_{ij}}(e_j) &= 0, \\ (\operatorname{ad} f_i)^{1-a_{ij}}(f_j) &= 0. \end{align*}