Proposition 2

(i) The Lie algebra $\mathfrak{a}_+$ is generated by the family $(x_{\alpha})_{\alpha\in B}$.

(ii) The Lie algebra $\mathfrak{a}_-$ is generated by the family $(x_{-\alpha})_{\alpha\in B}$.

(iii) The family $(h_{\alpha})_{\alpha\in B}$ is a basis of the vector space $\mathfrak{a}^0$.

(iv) The vector space $\mathfrak{a}$ is the direct sum of $\mathfrak{a}_+,\mathfrak{a}^0,\mathfrak{a}_-$.

Let $\mathfrak{r}$ (resp. $\mathfrak{n}$) be the Lie subalgebra of $\mathfrak{a}$ generated by $(x_{\alpha})_{\alpha\in B}$ (resp. $(x_{-\alpha})_{\alpha\in B}$), and $\mathfrak{h}$ the vector subspace of $\mathfrak{a}$ generated by $(h_{\alpha})_{\alpha\in B}$. Since the $x_{\alpha}$ are homogeneous elements of $\mathfrak{a}_+,\mathfrak{r}$ is a graded subalgebra of $\mathfrak{a}_+$; hence, $[\mathfrak{h},\mathfrak{r}]\subset \mathfrak{r}$, so $\mathfrak{h}+\mathfrak{r}$ is a subalgebra of $\mathfrak{a}$; since

$[x_{-\alpha},\mathfrak{r}]\subset \mathfrak{h}+\mathfrak{r}$ for all $\alpha \in B$. Similarly, $\mathfrak{n}$ is a graded subalgebra of $\mathfrak{a}_-$, one has $[\mathfrak{h},\mathfrak{n}]\subset \mathfrak{n},\mathfrak{h}+\mathfrak{n}$ is a subalgebra of $\mathfrak{n}$, and $[x_{\alpha},\mathfrak{n}]\subset \mathfrak{h}+\mathfrak{n}$ for all $\alpha \in B$. Put $\mathfrak{a}'=\mathfrak{r}+\mathfrak{h}+\mathfrak{n}$. The preceding shows that $\mathfrak{a}'$ is stable under ad $x_{\alpha}$, ad $h_{\alpha}$ and ad $x_{-\alpha}$ for all $\alpha \in B$, and hence is an ideal of $\mathfrak{a}$. Since $\mathfrak{a}'$ contains $x_{\alpha}, h_{\alpha}, x_{-\alpha}$ for all $\alpha \in B,\mathfrak{a}'=\mathfrak{a}$. It follows from this that the inclusions $\mathfrak{r}\subset \mathfrak{a}_+,\mathfrak{h}\subset \mathfrak{a}^0$, $\mathfrak{n}\subset \mathfrak{a}_-$ are equalities, which proves the proposition.