  This paper contains a classification of smooth Kaluza--Klein reductions (by
one-parameter subgroups) of the maximally supersymmetric anti de Sitter
backgrounds of supergravity theories. We present a classification of
one-parameter subgroups of isometries of anti de Sitter spaces, discuss the
causal properties of their orbits on these manifolds, and discuss their action
on the space of Killing spinors. We analyse the problem of which quotients
admit a spin structure. We then apply these results to write down the list of
smooth everywhere spacelike supersymmetric quotients of AdS_3 x S^3 (x R^4),
AdS_4 x S^7, AdS_5 x S^5 and AdS_7 x S^4, and the fraction of supersymmetry
preserved by each quotient. The results are summarised in tables which should
be useful on their own. The paper also includes a discussion of supersymmetry
of singular quotients.
