Strongly Doubly Reversible Pairs in Quaternionic Unitary Group of Signature $(n,1)$

Let $\text{PSp}(n,1)$ denote the isometry group of the quaternionic hyperbolic space ${\mathbb{H}}^n$. A pair $(g_1,g_2)$ $\text{PSp}(n,1)$ is \emph{strongly doubly reversible} if $(g_1,g_2)$ and $(g_1^{-1},g_2^{-1})$ are simultaneously conjugate in $\text{PSp}(n,1)$ by an involution. Equivalently, there exist involutions $i_1,i_2,i_3\in \text{PSp}(n,1)$ such that $g_1=i_1{i}_2$, $g_2=i_1{i}_3$. We prove that the set of such pairs has Haar measure zero in $\text{PSp}(n,1)\times \text{PSp}(n,1)$. The same result also holds for $\text{PSp}(n)\times \text{PSp}(n)$ for $n\ge 2$. In the special case $n=1$, we show that every pair of elements in $\text{PSp}(1)$ is strongly doubly reversible. Applying this result, we give a shorter proof of a theorem of Basmajian and Maskit showing that every pair of elements in $\mathrm{S}\mathrm{O}(4)$ is strongly doubly reversible. Furthermore, we derive the necessary conditions for a pair of hyperbolic elements in $\text{PSp}(1,1)$ to be strongly doubly reversible and provide a quantitative characterization of such pairs.