Remark 1

By Prop. 3 of §2, no. 4, the manifold $(G/T)\times A$ is simply-connected. It follows that $\varphi_A$ is a universal covering of $G_r$; since $\pi_1(G_r)$ is canonically isomorphic to $\pi_1(G)$ (no. 1, Prop. 1 and General Topology, Chap. XI, in preparation), we recover the fact that $\pi_1(G)$ can be identified with $H_A$ (that is with $\Gamma (T)/N(G,T)$).