of $H\times K$ on $G$ given by $$\label{hgk}(h,k)\cdot g=hgk^{-1},$$ is Cohen-Macaulay.

For the equivalence between (b) and (c) observe that we have the ring isomorphisms $$H^*_{H\times K}(G)\simeq H^*_H(G/K),$$ due to the fact that the factor $K$ of the product $H\times K$ acts freely on $G$. The equivalence between (a) and (c) can be proved in a similar way, although this time one also needs to take into account that the $H\times K$-action in the lemma is equivalent to the group action $(H\times K)\times G \to G$, $(h, k)\cdot g = kgh^{-1}.$ The result now follows from Cor. \[cmcrit\].

In the light of item (c) above, the following proposition is useful. It is a direct consequence of a certain Sullivan model obtained by J. D. Carlson and C.-K. Fok in [@Ca-Fo Sect. 3.1] (see also [@Ca1 Sect. 8.8.3] or [@Ca3 end of Sect. 2]), which describes the equivariant cohomology of actions on $G$ of type (\[hgk\]), the acting group being a general subgroup of $G\times G$. Denote by ${\mathfrak g}, {\mathfrak k}, {\mathfrak h}$ the Lie algebras of $G$, $K$, and $H$.

[(J. D. Carlson and C.-K. Fok)]{} \[carlson\] If $K$ and $H$ are connected, then the equivariant cohomology of the 