In $\mathcal N = 2$ Supergravity the scalar components of Hypermultiplets form a quaternionic Kaehler manifold. Only isometries of this so-called target manifold can be gauged.
I am interested in gauging an $SO(10)$ group and found that $$ \frac{SO(n,4)}{SO(n) \times SO(4)}$$ is a quaternionic Kaehler manifold (in a classification due to Wolf, I think).
As a physicist, I need rather explicit expressions. In the literature I find constructions of all the important quantities, starting from an explicit expression for the coset representative. These constructions are usually done for the simple case of the quaternionic Kaehler manifold $Sp(1,1)/(Sp(1) \times Sp(1))$. Even there, the form of the coset representative is never derived or motivated, just given.
I expect there is a "recipe" for constructing an explicit representation for a coset representative, but going through the physics publications I have not found any.
How can I construct an explicit expression for a representative of the coset given above? Is there a general procedure to construct such a quantity?
Edit: As an immediate followup, I am interested in the spinor $\mathbf{16}$ representation of $SO(10)$. Are there pitfalls to avoid in that case?
Let $G$ be one of $\mathrm{O}(p,q)$, $\mathrm{U}(p,q)$, or $\mathrm{Sp}(p,q)$, or at least their identity components, and let $\mathfrak{g}$ be the corresponding lie algebra. Every element of $\mathfrak{g}$ is of the block form
$$ \begin{bmatrix} A & X^\dagger \\ X & B \end{bmatrix} $$
where $A$ and $B$ are skew-adjoint. The identity component $H$ of the block-diagonal elements of $G$, i.e. the identity component of $O(p)\times O(q)$, $U(p)\times U(q)$, or $\mathrm{Sp}(p)\times\mathrm{Sp}(q)$, has lie algebra consisting of the same matrices but with $X=0$. Exponentiating the complementary subspace should yield a transversal for $H$, at least up to $\pi_0$. This is parametrized by all matrices $X$ via
$$ \exp\begin{bmatrix} 0 & X^\dagger \\ X & 0\end{bmatrix} = \begin{bmatrix} \cosh(\sqrt{X^\dagger X}) & X^\dagger\mathrm{sinch}(\sqrt{X^\dagger X}) \\ X\mathrm{sinch}(\sqrt{XX^\dagger}) & \cosh(\sqrt{XX^\dagger}) \end{bmatrix}$$
where $\mathrm{sinch}(x)=\sinh(x)/x$.