Let $A=(a_{ij})_{1 \le i,j \le n}$ be an indecomposable generalized Cartan matrix of finite type. Let $\mathfrak g(A)$ be the complex Lie algebra generated by elements $e_1,\dots,e_n,f_1,\dots,f_n,h_1,\dots,h_n$ subject to the Chevalley-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,\\ (\operatorname{ad} e_i)^{1-a_{ij}}(e_j)&=0 \quad \text{for } i \ne j,\\ (\operatorname{ad} f_i)^{1-a_{ij}}(f_j)&=0 \quad \text{for } i \ne j. \end{align*}

Let $\mathfrak h \subset \mathfrak g(A)$ be the subspace spanned by the images of $h_1,\dots,h_n$. Then $\mathfrak g(A)$ is a finite-dimensional complex semisimple Lie algebra, $\mathfrak h$ is a Cartan subalgebra of $\mathfrak g(A)$, and the root system of $\mathfrak g(A)$ relative to $\mathfrak h$ is the finite reduced root system whose Cartan matrix, with respect to the simple roots dual to $h_1,\dots,h_n$, is $A$.