Prop. \[crit\], we may assume that ${\rm rank} \ \!  K \le  {\rm rank} \ \! H$. Thus, by Lemma \[torus\], there exists $T\subset K$ a maximal torus and $g_0\in G$ such that $g_0Tg_0^{-1}\subset H$. The actions $K \times G/H\to G/H$ and $g_0Kg_0^{-1} \times G/H\to G/H$ given by left translation are equivalent, relative to the map $$(K, G/H)  \to (g_0Kg_0^{-1}, G/H), \quad (k, gH) \mapsto (g_0kg_0^{-1}, g_0gH).$$ Thus it is sufficient to show that the $g_0Kg_0^{-1}$-action on $G/H$ is Cohen-Macaulay. Equivalently, by means of Prop. \[maxtor\] above, that the action of $g_0Tg_0^{-1}$ on $G/H$ is so. But this is clear, because, by the main result in [@Go], the action of $H$ on $G/H$ is equivariantly formal.

If $G={\rm Spin}(8)$, then, as in the proof of Lemma \[torus\], we can assume that both $\sigma$ and $\tau$ are outer automorphisms. It is known that the group of outer automorphisms of ${\rm Spin}(8)$ is isomorphic to the symmetric group $\Sigma_3$ on three letters and the fixed point set of any order-two outer automorphism is isomorphic to ${\rm Spin}(7)$, giving rise to a copy of the symmetric space ${\rm Spin}(8)/{\rm Spin}(7) = S^7$ (cf. [@Ad1 Sect. 2]). Thus Thm. \[main\] is a consequence of 