TITLE: Lie groups acting transitively (and isometrically) on anti de Sitter spaces
QUESTION [7 upvotes]: 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.

REPLY [2 votes]: In my paper 
Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups. Rend. Circ. Mat. Palermo (2) Suppl. No. 79 (2006), 87–97.
I classify transitive group of isometries of the real Hyperbolic space. 
These groups are $SO(n,1)$ and subgroups of 
$$Sim(n-1)=(\mathbb{R}_+\times SO(n-1))\times \mathbb{R}^{n-1}$$ 
that preserve an isotropic line in the Minkowski space and do not preserve any non-degenerate subspace of the Minkowski space (weakly-irreducible subgroups), in addition they have non-trivial projection on $\mathbb{R}_+$. 
The connected group of isometries of AdS_n is $SO(n-1,2)$, hence, if a connected $G$ acts isometrically, it is contained in  $SO(n-1,2)$. From the transitivity it follows that the representation is weakly irreducible. Some of weakly irreducible subgroups of $SO(n-1,2)$ can be found in 
Ikemakhen, A. Sur l'holonomie des variétés pseudo-riemanniennes de signature (2,2+n). (French) [Holonomy of pseudo-Riemannian manifolds with signature (2,2+n)] Publ. Mat. 43 (1999), no. 1, 55–84. 
I think that it is possible to classify all transitively acting subgroups.