Proposition 4

Let $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ be a framed semi-simple Lie algebra. Let $(n(\alpha , \beta ))_{\alpha ,\beta\in B}$ and $(X_{\alpha})_{\alpha\in B\cup(-B)}$ be the corresponding Cartan matrix and generating family.

(i) The family $((X_{\alpha}, H_{\alpha}, X_{-\alpha}))_{\alpha\in B}$ and the relations (16) to (22) of no. 3 constitute a presentation of $\mathfrak{g}$.

(ii) The family $(X_{\alpha})_{\alpha\in B}$ and the relations (21) of no. 3 constitute a presentation of the subalgebra of $\mathfrak{g}$ generated by $(X_{\alpha})_{\alpha\in B}$.

Let R be the root system of $(\mathfrak{g},\mathfrak{h})$. Applying to R and B the constructions of nos. 2 and 3, we obtain objects that we shall denote by $\mathfrak{a}',\mathfrak{g}', X'_{\alpha}, H'_{\alpha}, . .$. instead of $\mathfrak{a},\mathfrak{g}, X_{\alpha}, H_{\alpha}, . .$..

There exists a homomorphism $\varphi$ from the Lie algebra $\mathfrak{g}'$ to the Lie algebra $\mathfrak{g}$ such that $\varphi (X'_{\alpha}) =X_{\alpha},\varphi (H'_{\alpha}) =H_{\alpha},\varphi (X'_{-\alpha}) =X_{-\alpha}$ for all $\alpha \in B$ (Prop. 1). Since dim $\mathfrak{g}'$ = Card R + Card B = dim$\mathfrak{g},\varphi$ is bijective. This proves (i).

The subalgebra of $\mathfrak{g}'=\mathfrak{a}'/(\mathfrak{n}'\oplus \theta '\mathfrak{n}') = (\mathfrak{a}'_+\oplus {\mathfrak{a}'}^0\oplus \mathfrak{a}'_-)/(\mathfrak{n}'\oplus \theta '\mathfrak{n}')$ generated by $(X'_{\alpha})_{\alpha\in B}$ can be identified with $\mathfrak{a}'_+/\mathfrak{n}'$. In view of Prop. 3 and the definition of $\mathfrak{n}'$, this proves (ii).