  Using a reconstruction theorem, we prove that the supersymmetry conditions
for a certain class of flux backgrounds are equivalent with a tractable
subsystem of relations on differential forms which encodes the full set of
contraints arising fom Fierz identities and from the differential and algebraic
conditions on the internal part of the supersymmetry generators. The result
makes use of the formulation of such problems through K\"{a}hler-Atiyah
bundles, which we developed in previous work. Applying this to the most general
${\cal N}=2$ flux compactifications of 11-dimensional supergravity on
8-manifolds, we can extract the conditions constraining such backgrounds and
give an overview of the resulting geometry, which generalizes that of
Calabi-Yau fourfolds.
