Proposition 1

Put $n=$ dim G. There exists a compact real-analytic manifold V of dimension $n-3$ and an analytic map $\varphi : V\rightarrow G$ whose image is G -- $G_r$.

Let $\alpha \in R(G,T)$; put $V_{\alpha}= (G/$Z(Ker$\alpha$ ))$\times$ (Ker$\alpha$ ), and let $\varphi_{\alpha}$ be the morphism from $V_{\alpha}$ to G such that, for all $g\in G$ and all $t\in$ Ker $\alpha$, we have $\varphi_{\alpha}(\overline{g}, t) =gtg^{-1}$ (we denote by $\overline{g}$ the coset of $g$ modulo Z(Ker$\alpha$ )). Then $V_{\alpha}$ is a compact real-analytic manifold of dimension

dim $V_{\alpha}=$ dim $G-$ dim Z(Ker $\alpha$ ) $+$ dim Ker$\alpha$

$=n-$ (dim T + 2) + (dim $T-1$) $=n-3$

(§4, no. 5, Th. $1$)$;\varphi_{\alpha}$ is a morphism of real-analytic manifolds, and the image of $\varphi_{\alpha}$ consists of the elements of G conjugate to an element of Ker $\alpha$. It now suffices to take V to be the sum of the manifolds $V_{\alpha}$, and $\varphi$ to be the morphism inducing $\varphi_{\alpha}$ on each $V_{\alpha}$.