Lie group $G$ is compact, connected, and semisimple (the case when $G$ is not necessarily semisimple is discussed in Rem. \[nonsemi\]). Due to Propositions \[crit\] and \[redcon\], we only need to show that the equivariant cohomology ring of the $H_0 \times K$-action on $G$ is Cohen-Macaulay. By Prop. \[carlson\], we lose no generality by assuming that $G$ is simply connected, since otherwise we can replace the triple $G$, $H_0$, $K$ with the universal cover of $G$ along with its two connected and closed subgroups that are locally isomorphic to $H_0$ and $K$ respectively. Under the aforementioned assumption, the fixed point sets of $\sigma$ and $\tau$ are connected, see [@He Thm. 1.8, Ch. VII], and thus equal to $H_0$ and $K$ respectively.

The case when $G$ is simple {#sec:simple}
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We will prove Thm. \[main\] under the assumption that $G$ is simple and simply connected, which is valid throughout the whole subsection. The induced automorphisms of $\g$ will also be denoted by $\s$ and $\t$. In general, we will not make any notational distinction between an automorphism of $G$ and the induced Lie algebra automorphism.

\[torus\] Assume $G$ is not equal to ${\rm Spin}(8)$. If ${\rm rank} \ \!  K \le  