We construct the non-linear Kaluza-Klein ansätze 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 ansätze to embed the charged AdS5, AdS4 and AdS7 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 transverse dimensions, in their near-horizon decoupling limits. The horizons of the black holes coincide with the world-volumes of the branes. The Kaluza-Klein ansätze also allow the black holes with spherical or hyperbolic horizons to be reinterpreted in D = 10 or D = 11.