  Several recent papers have made considerable progress in proving the
existence of remarkable consistent Kaluza-Klein sphere reductions of D=10 and
D=11 supergravities, to give gauged supergravities in lower dimensions. A proof
of the consistency of the full gauged SO(8) reduction on S^7 from D=11 was
given many years ago, but from a practical viewpoint a reduction to a smaller
subset of the fields can be more manageable, for the purposes of lifting
lower-dimensional solutions back to the higher dimension. The major complexity
of the spherical reduction Ansatze comes from the spin-0 fields, and of these,
it is the pseudoscalars that are the most difficult to handle. In this paper we
address this problem in two cases. One arises in a truncation of SO(8) gauged
supergravity in four dimensions to U(1)^4, where there are three pairs of
dilatons and axions in the scalar sector. The other example involves the
truncation of SO(6) gauged supergravity in D=5 to a subsector containing a
scalar and a pseudoscalar field, with a potential that admits a second
supersymmetric vacuum aside from the maximally-supersymmetric one. We briefly
discuss the use of these emdedding Ansatze for the lifting of solutions back to
the higher dimension.
