  We study the spinorial Killing equation of supergravity involving a torsion
3-form $\T$ as well as a flux 4-form $\F$. In dimension seven, we construct
explicit families of compact solutions out of 3-Sasakian geometries, nearly
parallel $\G_2$-geometries and on the homogeneous Aloff-Wallach space. The
constraint $\F \cdot \Psi = 0$ defines a non empty subfamily of solutions. We
investigate the constraint $\T \cdot \Psi = 0$, too, and show that it singles
out a very special choice of numerical parameters in the Killing equation,
which can also be justified geometrically.
