  We prove the instability of some families of Riemannian manifolds with
non-trivial real Killing spinors. These include the invariant Einstein metrics
on the Aloff-Wallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are
all nearly ${\rm G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles
over certain irreducible Hermitian symmetric spaces. We also prove the
instability of most of the simply connected non-symmetric compact homogeneous
Einstein spaces of dimensions $5, 6, $ and $7$, including the strict nearly
K\"ahler ones (except ${\rm G}_2/{\rm SU}(3)$).
