Lemma 3

Let $B_0$ be a basis of $R(G,T)$. The group Int$_G(T)$ operates simply-transitively on the set of framings of $(G,T)$ of the form $(B_0,(U_{\alpha})_{\alpha\in B_0})$.

For all $\alpha \in B_0$, denote by $K(\alpha )$ the restriction of the quadratic form K to $V(\alpha )$; the operation of T on $V(\alpha )$ defines a morphism $\iota_{\alpha}: T\rightarrow \mathbf{S}\mathbf{O}(K(\alpha ))$. We have seen in no. 4 that $\mathbf{S}\mathbf{O}(K(\alpha ))$ can be identified with $\mathbf{U}$ in such a way that $\iota_{\alpha}$ is identified with the root $\alpha$. Since $B_0$ is a basis of R, it is a basis of the $\mathbf{Z}$-module Q(R) generated by the roots, hence a basis of the submodule $X(T/C(G))$ of X(T). It follows that the product of the morphisms $\iota_{\alpha}$ induces an isomorphism from $T/C(G)$ to the product of the groups $\mathbf{S}\mathbf{O}(K(\alpha ))$. But the latter group operates simply-transitively on the set of framings of $(G,T)$ whose first component is $B_0$.