Homogeneous spaces with vanishing Steenrod squaring operations

If $G$ is a compact, connected Lie group, $H$ is a closed subgroup of $G$ and $G/H$ has no nonzero Steenrod operations, then $G/H$ splits as a product of homogeneous spaces of simple Lie groups (the factors of $G$). This fact is used to classify transitive actions on spaces with vanishing Steenrod operations, namely product of certain Stiefel manifolds and spheres.