Killing spinors in supergravity with 4-fluxes

We study the spinorial Killing equation of supergravity involving a torsion 3-form $\text{T}$ as well as a flux 4-form $\text{F}$. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel ${\text{G}}_2$-geometries and on the homogeneous Aloff-Wallach space. The constraint $\text{F}\cdot \mathrm{\Psi }=0$ defines a non empty subfamily of solutions. We investigate the constraint $\text{T}\cdot \mathrm{\Psi }=0$, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically.