Topological Properties of Ad-semisimple Conjugacy Classes in Lie Groups

We prove that every connected component of the zero locus in a connected Lie group G of any real polynomial without multiple roots is a conjugacy class. As applications, we prove that any Ad-semisimple conjugacy class C of G is a closed embedded submanifold, and that for any connected subgroup H of G, every connected component of C boolean AND H is a conjugacy class of H. Corresponding results for adjoint orbits in real Lie algebras are also proved.