A recent study of filtered deformations of (graded subalgebras of) the
minimal five-dimensional Poincar\’e superalgebra resulted in two classes of
maximally supersymmetric spacetimes. One class are the well-known maximally
supersymmetric backgrounds of minimal five-dimensional supergravity, whereas
the other class does not seem to be related to supergravity. This paper is a
study of the Kaluza–Klein reductions to four dimensions of this latter class
of maximally supersymmetric spacetimes. We classify the lorentzian and
riemannian Kaluza–Klein reductions of these backgrounds, determine the
fraction of the supersymmetry preserved under the reduction and in most cases
determine explicitly the geometry of the four-dimensional quotient. Among the
many supersymmetric quotients found, we highlight a number of novel
non-homogeneous four-dimensional lorentzian spacetimes admitting $N=1$
supersymmetry, whose supersymmetry algebra is not a filtered deformation of any
graded subalgebra of the four-dimensional $N=1$ Poincar\’e superalgebra. Any of
these four-dimensional lorentzian spacetimes may serve as the arena for the
construction of new rigidly supersymmetric field theories.

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