hypothesis mentioned above, $G/K$ is simply connected as well, hence the $H$-action on $G/K$ is not only locally conjugate to one of the types (i)-(iv), as the result in [@Ko] says, but genuinely conjugate.

\(b) We also wish to explain the meaning of the number $n$ in the proposition above. It comes from the structure of $G/K$. For example, in case (i) the pair $(G, K)$ is equal to $(L \times \cdots \times L, \Delta(L)\times \cdots \times 
\Delta(L))$, where the first direct product has $2n$ factors, the second has $n$, and $\Delta(L) :=\{(g, g) | g\in L\}$. The cases (ii)-(iv) are left as an exercise to the reader.

In view of Cor. \[dirprod\], to prove Thm. \[main\] for $G$ semisimple it is sufficient to show that the actions (i)-(iv) are Cohen-Macaulay.

Let us start with (i). We use an inductive argument. Note that the first factor $L$ in $H'\times L^{n-1}\times K'$ acts freely on $L^{n}$, the orbit space being diffeomorphic to $L^{n-1}$ via $$L^{n}/L \to L^{n-1}, \quad (x_1, x_2, \ldots,  x_{n}) \mapsto (x_1 x_2, x_3 \ldots,  x_{n}).$$ Thus it is sufficient to prove that the action of $H'\times L^{n-2}\times K'$ on $L^{n-1}$ given by $$\begin{aligned}
{}&(h, g_2, \ldots, g_{n-1}, k)\cdot(x_2, 