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

Let (n,1) denote the isometry group of the quaternionic hyperbolic space Hⁿ. A pair (g₁,g₂) (n,1) is strongly doubly reversible if (g₁,g₂) and (g₁⁻¹,g₂⁻¹) are simultaneously conjugate in (n,1) by an involution. Equivalently, there exist involutions i₁,i₂,i₃ ∈ (n,1) such that g₁ = i₁ i₂, g₂ = i₁ i₃. We prove that the set of such pairs has Haar measure zero in (n,1) × (n,1). The same result also holds for (n) × (n) for n≥ 2. In the special case $n=1$, we show that every pair of elements in (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 SO(4) is strongly doubly reversible. Furthermore, we derive the necessary conditions for a pair of hyperbolic elements in (1,1) to be strongly doubly reversible and provide a quantitative characterization of such pairs.