Motivated by the search for new gravity duals to M2 branes with N>4
supersymmetry --- equivalently, M-theory backgrounds with Killing superalgebra
osp(N∣4) for N>4 --- we classify homogeneous M-theory
backgrounds with symmetry Lie algebra so(n)⊕so(3,2) for n=5,6,7. We find that there are no new backgrounds
with n=6,7 but we do find a number of new (to us) backgrounds with n=5. All
backgrounds are metrically products of the form AdS4×P7, with P riemannian and homogeneous under the action of
SO(5), or S4×Q7 with Q lorentzian and homogeneous
under the action of SO(3,2). At least one of the new
backgrounds is supersymmetric (albeit with only N=2) and we show that it can
be constructed from a supersymmetric Freund--Rubin background via a Wick
rotation. Two of the new backgrounds have only been approximated numerically.
(The second version of this paper includes an appendix by Alexander~S.~Haupt,
closing a gap in our original analysis.)Comment: 56 page