Definition 1

Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra, R its root system. A framing of $(\mathfrak{g},\mathfrak{h})$ is a pair $(B,(X_{\alpha})_{\alpha\in B})$, where B is a basis of R, and where, for all $\alpha \in B,X_{\alpha}$ is a non-zero element of $\mathfrak{g}^{\alpha}$. A framed semi-simple Lie algebra is a sequence $(\mathfrak{g},\mathfrak{h},B,(X_{\alpha})_{\alpha\in B})$ where $(\mathfrak{g},\mathfrak{h})$ is a split semi-simple Lie algebra, and where $(B,(X_{\alpha})_{\alpha\in B})$ is a framing of $(\mathfrak{g},\mathfrak{h})$.

A framing of $\mathfrak{g}$ is a framing of $(\mathfrak{g},\mathfrak{h})$, where $\mathfrak{h}$ is a splitting Cartan subalgebra of $\mathfrak{g}$.

Let $a_1= (\mathfrak{g}_1,\mathfrak{h}_1,B_1,(X_{\alpha}^1)_{\alpha\in B_1})$ and $a_2= (\mathfrak{g}_2,\mathfrak{h}_2,B_2,(X_{\alpha}^2)_{\alpha\in B_2})$ be framed semi-simple Lie algebras. An isomorphism from $a_1$ to $a_2$ is an isomorphism $\varphi$ from $\mathfrak{g}_1$ to $\mathfrak{g}_2$ that takes $\mathfrak{h}_1$ to $\mathfrak{h}_2, B_1$ to $B_2$, and $X_{\alpha}^1$ to $X_{\psi \alpha}^2$ for all $\alpha \in B_1$ (where $\psi$ is the contragredient map of $\varphi |\mathfrak{h}_1$). In this case, $\varphi$ is said to transform the framing $(B_1,(X_{\alpha}^1)_{\alpha\in B_1})$ to the framing $(B_2,(X_{\alpha}^2)_{\alpha\in B_2})$.

If $(B,(X_{\alpha})_{\alpha\in B})$ is a framing of $(\mathfrak{g},\mathfrak{h})$, there exists, for all $\alpha \in B$, a unique element $X_{-\alpha}$ of $\mathfrak{g}^{-\alpha}$ such that $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}($§2, no. 2, Th. 1 (iv)). The family $(X_{\alpha})_{\alpha\in B\cup(-B)}$ is called the generating family defined by the framing (cf. Prop.1). This is also the generating family defined by the framing $(-B,(X_{\alpha})_{\alpha\in -B})$. For all $\alpha \in B\cup (-B)$, let $t_{\alpha}\in k^*$, and assume that $t_{\alpha}t_{-\alpha}= 1$ for all $\alpha \in B$. Then $(t_{\alpha}X_{\alpha})_{\alpha\in B\cup(-B)}$ is the generating family defined by the framing $(B,(t_{\alpha}X_{\alpha})_{\alpha\in B})$.