  For a split semisimple Chevalley group scheme G with Lie algebra g over an
arbitrary base scheme S, we consider the quotient of g by the adjoint action of
G. We study in detail the structure of g over S. Given a maximal torus T with
Lie algebra t and associated Weyl group W, we show that the Chevalley morphism
t/W -> g/G is an isomorphism except for the group Sp_{2n} over a base with
2-torsion. In this case this morphism is only dominant and we compute it
explicitly. We compute the adjoint quotient in some other classical cases,
yielding examples where the formation of the quotient g -> g/G commutes, or
does not commute, with base change on S.
