Theorem 1

Let R be a reduced root system, B a basis of R. Let $\mathfrak{g}$ be the Lie algebra defined by the generating family $((X_{\alpha}, H_{\alpha}, X_{-\alpha}))_{\alpha\in B}$ and the relations (16) to (22). Let $\mathfrak{h}=\sum_{\alpha\in B}kH_{\alpha}$. Then $(\mathfrak{g},\mathfrak{h})$ is a split semi-simple Lie algebra. The isomorphism $\mu \rightarrow \mu_B$ from V to $\mathfrak{h}^*$ maps R to the root system of $(\mathfrak{g},\mathfrak{h})$. For all $\mu\in R,\mathfrak{g}^\mu$ is the eigenspace relative to the root $\mu$.

The proof follows that of Lemmas 8, 9, 10, 11.