I hope this question is not deemed too localised.

Recall that *anti de Sitter space* is the lorentzian analogue of hyperbolic space; that is, a simply-connected lorentzian manifold of constant negative sectional curvature. More precisely, the $n$-dimensional anti de Sitter space $\mathrm{AdS}_n$ is the universal cover of the one-sheeted hyperboloid in $\mathbb{R}^{n+1}$ cut out by the following quadric:
$$
x_1^2 + x_2^2 + \dots + x_{n-1}^2 - x_n^2 - x_{n+1}^2 = - R^2
$$
where $R>0$ is the radius of curvature. The ambient metric is flat with signature $(n-1,2)$ and the induced metric on the hyperboloid has constant negative sectional curvature proportional to $1/R$.

The group $\mathrm{SO}(n-1,2)$ acts transitively and isometric on the hyperboloid; indeed, the hyperboloid is diffeomorphic to $\mathrm{SO}(n-1,2)/\mathrm{SO}(n-1,1)$.

This is analogous to the better known fact that the round $n$-sphere is diffeomorphic to $\mathrm{SO}(n+1)/\mathrm{SO}(n)$. However, it happens that for some spheres, a proper subgroup $G \subset \mathrm{SO}(n+1)$ already acts transitively on the sphere and in some cases such $G$ is (locally) isomorphic to $\mathrm{SO}(p)$ for some $p\lt n+1$. A case in point is the round 7-sphere, on which $\mathrm{Spin}(6)$ acts transitively with stabiliser subgroup $\mathrm{SU}(3)$.

I am interested in knowing whether something like that can happen for anti de Sitter space. In particular, I would like to know whether $\mathrm{SO}(n-1,2)$ (or some other group with isomorphic Lie algebra) can act transitively on $\mathrm{AdS}_p$ for some $p\gt n$. We already know it acts on $\mathrm{AdS}_n$ and it cannot act (effectively) on $\mathrm{AdS}_p$ for $p\lt n$ by dimension. As in the case of the spheres, this can perhaps only happen for low values of $n$ and in fact, to be perfectly honest, I am presently only interested in $n=4$.

Hence let me ask the following

**Question**

Can $\mathrm{SO}(n-1,2)$ act locally transitively (and isometrically!) on $\mathrm{AdS}_p$ for some $p\gt n$? In particular, can this happen for $n=4$?

Thank you in advance.

**Epilogue**

I should have mentioned that I have checked by explicit calculation that $\mathrm{SO}(3,2)$ cannot act locally transitively and isometrically on $\mathrm{AdS}_5$, but before trying my hand at $\mathrm{AdS}_6$ or higher, I thought of asking here first.