Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a connected semi-simple real linear algebraic group, $K$ a maximal compact subgroup, which is the fixed part of a Cartan involution $\theta$ on $G$.

And by equivariant embeddings is understood a smooth embedding of symmetric domains: $i:D_1\rightarrow D$, with $D_1$ defined by a connected semi-simple subgroup $G_1\subset G$, stabilized by $\theta$. In this case $\theta_{G_1}$ is a Cartan involution on $G_1$, with fixed part $K_1$, and thus $D_1$ is isomorphic to $G_1/K_1$. If one regards $D=G/K$ as the set of conjugates of $\theta$ under $G$, then $D_1$ is the $G_1$-orbit in $G\theta=D$.

The question is to understand refinement of such embeddings, i.e. what kind of chain of equivariant embeddings can one get like $D_1\rightarrow D_2\rightarrow D$?

More specifically, consider the centralizer $Z=Z(G_1,G)$ of $G_1$ in $G$. Note that $Z$ is a connected subgroup. Two cases arise:

(1) $Z$ is non-compact; then one can show that $G_1$ extends to a non-trivial parabolic subgroup of $G$, i.e. $G_1\subset P\subsetneq G$.

(2) $Z$ is compact; then $G_1$ cannot be extended to a non-trivial parabolic, and any subgroup of $G$ containing $G_1$ has to be reductive.

for a reference of the characterization above, see "non-divergence of translates of certain algebraic measures", lemma 5.1, by A.Eskin, S.Mozes, N.Shah, in GAFA 7(1997), pp.48-80

Write $N^\circ=N^\circ(G_1,G)$ for the connected component of the normalizer of $G_1$ in $G$. Then this reductive subgroup is also stable under the Cartan involution $\theta$, and the corresponding symmetric subdomain given as the $N^\circ$-orbit of $\theta$ in $D=G\theta$.

If (2) happens, then $N^\circ$ gives the same symmetric subdomain $D_1$; otherwise in case (1) $N^\circ$ could give a larger symmeric subdomain $D_2$, in which $D_1$ serves as a factor $D_2=D_1\times D_1'$.

I would like to know

(i) how to characterize more geometrically the difference between (1) and (2);

(ii) is (2) transitive? namely, if one has a chain of equivariant embeddings $D_1\rightarrow D_2\rightarrow D$ given by semi-simple subgroups $G_1\subset G_2\subset G$ stable under a common Cartan involution, such that $Z(G_1,G_2)$ and $Z(G_2,G)$ both compact, can one show $Z(G_1,G)$ compact also?

(iii) does any difference occur if one works with equivariant embeddings of Hermitian symmetric domains?

Sorry for the lengthy presentation.