  We construct the non-linear Kaluza-Klein ans\"atze describing the embeddings
of the U(1)^3, U(1)^4 and U(1)^2 truncations of D=5, D=4 and D=7 gauged
supergravities into the type IIB string and M-theory. These enable one to
oxidise any associated lower dimensional solutions to D=10 or D=11. In
particular, we use these general ans\"atze to embed the charged AdS_5, AdS_4
and AdS_7 black hole solutions in ten and eleven dimensions. The charges for
the black holes with toroidal horizons may be interpreted as the angular
momenta of D3-branes, M2-branes and M5-branes spinning in the tranverse
dimensions, in their near-horizon decoupling limits. The horizons of the black
holes coincide with the worldvolumes of the branes. The Kaluza-Klein ans\"atze
also allow the black holes with spherical or hyperbolic horizons to be
reinterpreted in D=10 or D=11.
