  Let G be an n-dimensional semisimple compact and connected Lie group acting
on both the Lie algebra g of G and its dual g*. We show that a nondegenerate
Killing form of G induces an Ad*-equivariant isomorphism of g onto g* which, in
turn, induces by passage to quotients a symplectomorphism between adjoint and
coadjoint orbit spaces of G.
