(hx_1g_1^{-1}, g_1x_2g_2^{-1}, \ldots, g_{n-2}x_{n-1}g_{n-1}^{-1}, g_{n-1}x_n K'),\end{aligned}$$

-   the action of $L^{n-1}$ on $H'\backslash L\times L^{n-2}\times L/K'$ defined by $$\begin{aligned}
    {}&(g_1,\ldots, g_{n-1})\cdot (H'x_1, x_2, \ldots, x_{n-1}, x_n K')
    \\{}&=(H'x_1g_1^{-1}, g_1x_2g_2^{-1}, \ldots, g_{n-2}x_{n-1}g_{n-1}^{-1}, g_{n-1}x_n K'),\end{aligned}$$

-   the action of $L^n$ on $L^n$ defined by $$\begin{aligned}
    {}&(g_1, \ldots, g_n)\cdot(x_1,\ldots, x_n)\\{}&
    =(g_1x_1g_2^{-1}, g_2x_2g_3^{-1}, \ldots, g_{n-1}x_{n-1}g_n^{-1}, g_n x_n\alpha(g_1)^{-1}),\end{aligned}$$

where $L$ is a simply connected, simple and compact Lie group, $H',K'\subset L$ are fixed points of involutions of $L$, $\alpha$ is an outer or trivial automorphism of $L$, and $n$ is an arbitrary integer, at least equal to 1 in cases (i), (ii), and (iv), respectively to 2 in case (iii).

For the reader’s convenience, here are a few details concerning this result.

\(a) Proposition \[andreas\] above is just a consequence of the result actually proved in [@Ko]. Namely, we are here in the special case when $G$ is simply connected. Thus our $H$-action on $G/K$ is of the Hermann type in the sense of [@Ko Def. 15] (see the definition of locally symmetric subgroups at the beginning of [@Ko Sect. 2] and also recall the well-known fact that the identity component of the isometry group of the symmetric space $G/K$ is just $G$). Under the 