Corollary 3

Let B be a basis of R. The group Ker $\varepsilon =f(T_Q)$ operates simply-transitively on the set of framings of $(\mathfrak{g},\mathfrak{h})$ of the form $(B,(X_{\alpha})_{\alpha\in B})$.

This follows immediately from Prop. 2.

Let $\alpha \in R,X_{\alpha}\in \mathfrak{g}^{\alpha},X_{-\alpha}\in \mathfrak{g}^{-\alpha}$ be such that $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$. We have seen (§2, no. 2, Th. 2) that, for all $t\in k^*$, the restriction of the elementary automorphism

to $\mathfrak{h}$ is the transpose of $s_{\alpha}$; so $\varepsilon (\theta_{\alpha}(t)) =s_{\alpha}$ and consequently $\theta_{\alpha}(t)\theta_{\alpha}(-1)\in$ Ker $\varepsilon$.