  We prove that Ad-semisimple conjugacy classes in a connected Lie group $G$
are closed embedded submanifolds of $G$. We also prove that if $\alpha:H\to G$
is a homomorphism of connected Lie groups such that the kernel of $\alpha$ is
discrete in $H$, then for an Ad-semisimple conjugacy class $C$ in $G$, every
connected component of $\alpha^{-1}(C)$ is a conjugacy class in $H$.
Corresponding results for adjoint orbits in real Lie algebras are also proved.
