  Let G be a simple algebraic group over an algebraically closed field of
characteristic zero or positive odd, good characteristic. Let B be a Borel
subgroup of G. We show that the spherical conjugacy classes of G intersect only
the double cosets of B in G corresponding to involutions in the Weyl group of
G. This result is used to prove that for a spherical conjugacy class O with
dense B-orbit v_0 contained in BwB there holds l(w)+rk(1-w)=dim(O) extending a
characterization of spherical conjugacy classes obtained by N. Cantarini, M.
Costantini and the author to the case of groups over fields of odd, good
characteristic.
