Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the collection of these known as para-Hermitian symmetric spaces.
For more context, I am interested in the following problem. Let $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ be the grading associated to a self-dual symmetric R-space $M$ (i.e. a compact Hermitian symmetric space or a real form of one). That is $\mathfrak{p} = \mathfrak{g}_0 \oplus \mathfrak{g}_1$, $\mathfrak{q} = \mathfrak{g}_0 \oplus \mathfrak{g}_{-1}$ are complementary parabolic subalgebras which are conjugate to each other and $\mathfrak{g}_{\pm 1}$ are their abelian nilradicals. Note also that $ \mathfrak{g}_0 \oplus (\mathfrak{g}_{-1} \oplus \mathfrak{g}_1) $ is a symmetric decomposition corresponding to the space of complementary pairs of parabolic subalgebras (also known as a para-hermitian symmetric space) which I'll denote $Z$.
Assume first that $\mathfrak{g}$ is complex. Choosing a Cartan subalgebra contained in $\mathfrak{g}_0$ we get a root system which aligns with our decompositions. Then we choose a maximal set of strongly orthogonal roots $\{\beta_{1},\dots,\beta_{r}\}$ whose root spaces are in $\mathfrak{g}_1$ and use this to define a subalgebra $ \mathfrak{s} := \mathfrak{sl}_{\beta_1}\oplus \cdots \oplus \mathfrak{sl}_{\beta_r}$ ($r$ turns out to be the rank of $Z$). The set of Borel subalgebras of $\mathfrak{s}$ (itself a symmetric R-space diffeomorphic to $S^2 \times \cdots \times S^2$) can then be naturally embedded into $M$. In this case, $M$ is a (self-dual) Hermitian symmetric space and these are exactly the polyspheres of Harish-Chandra and it is not too hard to show there is only one of these up to conjugacy. I want to determine what this looks like for real symmetric R-spaces.
When we move to the real case, it becomes a little more complicated. Firstly, we replace $\mathfrak{s}$ by real forms (with no compact summands so we can meaningfully find parabolic subalgebras). These are precisely sums of $k$ copies of $\mathfrak{sl}(2,\mathbb{C})$ viewed as a real Lie algebra and $l$ copies of $\mathfrak{sl}(2,\mathbb{R})$:
$$ \mathfrak{s} = \bigoplus_{i=1}^k \mathfrak{sl}(2,\mathbb{C}) \oplus \bigoplus_{j=2k+1}^r \mathfrak{sl}(2,\mathbb{R}).$$
Of course, $k$ (or $l$) represents an invariant under conjugation but these can break down even further into conjugacy classes. For example, the projective quadric of signature $(p,q)$, $S^{p,q}$ has up to $3$ conjugacy classes (lower if $p$ or $q$ is less than $2$) corresponding to the inclusions $S^{2,0},S^{1,1},S^{0,2} \subset S^{p,q}$. Here $S^{2,0},S^{0,2}$ have $k=1$ and $S^{1,1}$ has $k=0$.
As an attempt to resolve this, I noted that each $\mathfrak{s}$ of type $(k,l)$ contains exactly $2^l$ Cartan subspaces of $Z$ up to conjugacy and moreover each Cartan subspace is contained in a unique $\mathfrak{s}$. So knowing how many conjugacy classes of Cartan subspaces of $Z$ there are would give some insight into how many conjugacy classes of these $\mathfrak{s}$ there are. I have tried to use papers such as "Orbits on affine symmetric spaces under the action of the isotropy subgroups" by Oshima and Matsuki to calculate these but I haven't been able to make any headway with that. In most treatments of para-Hermitian symmetric spaces I have found they define a single "maximally split" Cartan subspace (or at least its split part) but I am interested in all the possibilities.
