  Let $\mathcal{M}(n,m;\F \bp^n)$ be the configuration space of $m$-tuples of
pairwise distinct points in $\F \bp^n$, that is, the quotient of the set of
$m$-tuples of pairwise distinct points in $\F \bp^n$ with respect to the
diagonal action of ${\rm PU}(1,n;\F)$ equipped with the quotient topology. It
is an important problem in hyperbolic geometry to parameterize
$\mathcal{M}(n,m;\F \bp^n)$ and study the geometric and topological structures
on the associated parameter space. In this paper, by mainly using the
rotation-normalized and block-normalized algorithms, we construct the parameter
spaces of both $\mathcal{M}(n,m; \bhq)$ and $\mathcal{M}(n,m;\bp(V_+))$,
respectively.
