  Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$.
We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1)
\hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is
done by studying a natural non-associative algebra $\mathfrak{m}$ comming from
the affine structure of $Sp(1) \backslash H$. We compute the automorphism group
of $\mathfrak{m}$ and as a consecuence of this, we are able to compute the
isometry group of $Sp(1) \backslash H$ at least up to connected components.
Using these results, we obtain a uniqueness result on the structure of $Sp(1)
\backslash H$ together with an isometric left $Sp(k,l)$-action and classify its
finite volume quotients up to finite coverings. Finally, we classify arbitrary
isometric actions of $Sp(k,l)$ into connected, complete, analytic,
pseudo-Riemannian manifolds admitting a dense orbit of dimension bounded by
$\textrm{dim}(Sp(1) \backslash H)$.
