  Let X be the wonderful compactification of a semisimple adjoint algebraic
group. Extending the standard monomials on the flag variety, Chirivi and Maffei
constructed a basis of the space of global sections on X that is compatible
with all closures of GxG-orbits in X. We show that this basis is also
compatible with all BxB-orbit closures by defining subsets using only
combinatorics of the underlying paths. Furthermore, we construct standard
monomials on X that have properties similar to classical standard monomials.
