Proposition 8

Let $(X_{\alpha})_{\alpha\in R}$ be a Chevalley system for $(\mathfrak{g},\mathfrak{h})$. Let M be a $\mathbf{Z}$-submodule of $\mathfrak{h}$ containing the $H_{\alpha}$ and contained in the group of weights of $R^{\vee}$. Let $\mathfrak{g}_{\mathbf{Z}}$ be the $\mathbf{Z}$-submodule of $\mathfrak{g}$ generated by M and the $X_{\alpha}$. Then $\mathfrak{g}_{\mathbf{Z}}$ is a $\mathbf{Z}$-Lie subalgebra of $\mathfrak{g}$, and the canonical map from $\mathfrak{g}_{\mathbf{Z}}\otimes_{\mathbf{Z}}k$ to $\mathfrak{g}$ is an isomorphism.

If $\alpha , \beta \in R$ are such that $\alpha +\beta \in R$, then $N_{\alpha ,\beta}\in \mathbf{Z}$ (Prop. 7). On the other hand, if $\alpha \in R$ and $h\in M$, then $\alpha (h)\in \mathbf{Z}$ (Chap. VI, §1, no. 9). This proves that $\mathfrak{g}_{\mathbf{Z}}$ is a $\mathbf{Z}$-Lie subalgebra of $\mathfrak{g}$. On the other hand, M is a free abelian group of rank dim$\mathfrak{h}($Algebra, Chap. VII, §3, Th. 1), so $\mathfrak{g}_{\mathbf{Z}}$ is a free abelian group of rank dim$\mathfrak{g}$; this implies the last assertion.