Let $k$ be an algebraically closed field of characteristic $0$, let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over $k$, and let $\mathfrak h \subset \mathfrak g$ be a Cartan subalgebra with root system $\Phi \subset \mathfrak h^*$. Then there exist elements $e_\alpha \in \mathfrak g_\alpha$ for each $\alpha \in \Phi$ and a basis $h_1,\dots,h_\ell$ of $\mathfrak h$, where $\ell = \dim \mathfrak h$, such that $\{h_1,\dots,h_\ell\} \cup \{e_\alpha : \alpha \in \Phi\}$ is a basis of $\mathfrak g$ and all structure constants of $\mathfrak g$ in this basis are integers.

More explicitly, for all $\alpha,\beta \in \Phi$:

\begin{align*} [h_i,h_j] &= 0, \\ [h_i,e_\alpha] &= \alpha(h_i)e_\alpha \quad \text{with } \alpha(h_i) \in \mathbb Z, \\ [e_\alpha,e_{-\alpha}] &\in \bigoplus_{i=1}^{\ell} \mathbb Z h_i, \\ [e_\alpha,e_\beta] &= N_{\alpha,\beta}e_{\alpha+\beta} \end{align*}

whenever $\alpha+\beta \in \Phi$, with $N_{\alpha,\beta} \in \mathbb Z$, and

\begin{align*} [e_\alpha,e_\beta] = 0 \end{align*}

whenever $\alpha+\beta \notin \Phi$ and $\alpha \ne -\beta$.