Proposition 19

The group Int(G) operates simply-transitively on the set of framings of G.

Let $e= (T,B,(U_{\alpha}))$ and $e'= (T',B',(U'_{\alpha}))$ be two framings of G. There exist elements $g$ in G such that (Int $g$)$(T) = T'$, and these elements form a single coset modulo $N_G(T)$. Thus, we can assume that $T = T'$, and we must prove that there exists a unique element of Int$_G(N_G(T))$ that transforms $e$ to $e'$. By Chap. VI, §1, no. 5, Remark 4, there exists a unique element $w$ of W(R) such that $w(B) = B'$. Since W(R) can be identified with $N_G(T)/T$, there exists $n\in N_G(T)$ such that $w=$ Int $n$, and $n$ is uniquely determined modulo T. Thus, we can assume that $B = B'$, and we must prove that there exists a unique element of Int$_G(T)$ that transforms $e$ to $e'$, which is simply Lemma 3.