Lemma 2

There exists a family $(X_{\alpha})_{\alpha\in R}$ such that, for all $\alpha \in R$,

$X_{\alpha}\in \mathfrak{g}^{\alpha}$ and $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$.

Let $R_1$ be a subset of R such that $R = R_1\cup (-R_1)$ and $R_1\cap (-R_1) =\emptyset$. For $\alpha \in R_1$, choose an arbitrary non-zero element $X_{\alpha}$ of $\mathfrak{g}^{\alpha}$. There exists a unique $X_{-\alpha}\in \mathfrak{g}^{-\alpha}$ such that $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$ (Th. 1 (iv)). Then

If $(X_{\alpha})_{\alpha\in R}$ is one family satisfying the conditions of Lemma 2, the most general family satisfying these conditions is $(t_{\alpha}X_{\alpha})_{\alpha\in R}$ where $t_{\alpha}\in k^*$ and $t_{\alpha}t_{-\alpha}= 1$ for all $\alpha \in R$.

In the remainder of this number, we denote by $(X_{\alpha})_{\alpha\in R}$ a family satisfying the conditions of Lemma 2. We denote by $\langle \cdot ,\cdot \rangle$ a non-degenerate invariant symmetric bilinear form on $\mathfrak{g}$.

Every $x\in \mathfrak{g}$ can be written uniquely in the form

The bracket of two such elements can be calculated by means of the following formulas:

0 if $\alpha +\beta \notin R\cup \{0\}$

$[X_{\alpha}, X_{\beta}] =-H_{\alpha}$ if $\alpha +\beta = 0$

$N_{\alpha \beta}X_{\alpha+\beta}$ if $\alpha +\beta \in R$

the $N_{\alpha \beta}$ being non-zero elements of $k$.