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Abstract Through timelike dualities, one can generate exotic versions of M-theory with different spacetime signatures. These are the M *-theory with signature (9, 2, −), the M′-theory, with signature (6, 5, +) and the theories with reversed signatures (1, 10, −), (2, 9, +) and (5, 6, −). In (s, t, ±), s is the number of space directions, t the number of time directions, and ± refers to the sign of the kinetic term of the 3 form. The only irreducible pseudo-riemannian manifolds admitting absolute parallelism are, besides Lie groups, the seven-sphere S 7 ≡ SO(8)/SO(7) and its pseudo-riemannian version S 3,4 ≡ SO(4, 4)/SO(3, 4). [There is also the complexification SO 8 ℂ / SO 7 ℂ $$ \mathrm{SO}\left(8,\mathrm{\mathbb{C}}\right)/\mathrm{SO}\left(7,\mathrm{\mathbb{C}}\right) $$ , but it is of dimension too high for our considerations.] The seven-sphere S 7 ≡ S 7,0 has been found to play an important role in 11-dimensional supergravity, both through the Freund-Rubin solution and the Englert solution that uses its remarkable parallelizability to turn on non trivial internal fluxes. The spacetime manifold is in both cases AdS 4 × S 7. We show that S 3,4 enjoys a similar role in M ′-theory and construct the exotic form AdS 4 × S 3,4 of the Englert solution, with non zero internal fluxes turned on. There is no analogous solution in M ∗-theory.