Let $\mathbb S$ be the torus $\mathbb C^\times$ viewed as an algebraic group over $\mathbb R$. Let $G$ be any affine algebraic group over $\mathbb R$. The set $Hom(\mathbb S,G)$ of morphisms of real algebraic groups has a natural topology (e.g. choose an embedding $G\hookrightarrow GL_V$ and consider the topology given by the inclusion of this last set into a certain product of flag varieties...).

Question: is it easy to see that every connected component of $Hom(\mathbb S,G)$ is a $G(\mathbb R)^+$-conjugacy class? (here $G(\mathbb R)^+$ is the neutral connected component for the real topology)