  We present a simple class of warped-product vacuum (Ricci-flat) solutions to
ten and eleven-dimensional supergravity, where the internal space is flat and
the warp factor supports de Sitter (dS) and anti-de Sitter (AdS) vacua in
addition to trivial Minkowski vacua. We outline the construction of consistent
Kaluza-Klein (KK) reductions and show that, although our vacuum solutions are
non-supersymmetric, these are closely related to the bosonic part of well-known
maximally supersymmetric reductions on spheres. We comment on the stability of
our solutions, noting that (A)dS$_3$ vacua pass routine stability tests.
