  Consider an oriented four-dimensional Lorentzian manifold $(\widetilde{M}^{3,
1}, \widetilde{g})$ and an oriented seven-dimensional Riemannian manifold
$(M^{7}, g)$. We describe a class of decomposable eleven-dimensional
supergravity backgrounds on the product manifold $({\mathcal{M}}^{10,
1}=\widetilde{M}^{3,1} \times M^7, g_{{\mathcal{M}}}=\widetilde{g}+g)$, endowed
with a flux form given in terms of the volume form on $\widetilde{M}^{3, 1}$
and a closed $4$-form $F^{4}$ on $M^{7}$. We show that the Maxwell equation for
such a flux form can be read in terms of the co-closed 3-form
$\phi=\star_{7}F^{4}$. Moreover, the supergravity equation reduces to the
condition that $(\widetilde{M}^{3,1},\widetilde{g})$ is an Einstein manifold
with negative Einstein constant and $(M^7, g, F)$ is a Riemannian manifold
which satisfies the Einstein equation with a stress-energy tensor associated to
the 3-form $\phi$. Whenever this 3-form is generic, the Maxwell equation
induces a weak ${\rm G}_2$-structure on $M^{7}$ and then we obtain decomposable
supergravity backgrounds given by the product of a weak ${\rm G}_2$-manifold
$(M^7, \phi, g)$ with a Lorentzian Einstein manifold
$(\widetilde{M}^{3,1},\widetilde{g})$. We classify homogeneous 7-manifolds
$M^{7}=G/H$ of a compact Lie group $G$ and indicate the cosets which admit an
invariant or non-invariant ${\rm G}_2$-structure, or even no ${\rm
G}_2$-structure. Then we construct examples of compact homogeneous Riemannian
7-manifolds endowed with non-generic invariant 3-forms which satisfy the
Maxwell equation, but the construction of decomposable homogeneous supergravity
backgrounds of this type remains an open problem.
