Equivariant formality of isotropy actions on homogeneous spaces defined by Lie group automorphisms

We show that the isotropy action of a homogeneous space $G\mathrm{/}K$, where $G$ and $K$ are compact, connected Lie groups and $K$ is defined by an automorphism on $G$, is equivariantly formal and that $(G,K)$ is a Cartan pair.