Algebraic characterization of reversibility in the quaternionic Möbius group

An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic Möbius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that $$[A]\in \mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{H})\text{ is reversible}⟺{\beta }_A^2={\delta }_A^2,$$ where ${\beta }_A$ and ${\delta }_A$ are real conjugacy invariants associated with a lift $A\in \mathrm{S}\mathrm{L}(2,\mathbb{H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in $\mathrm{S}\mathrm{L}(2,\mathbb{H})$ and $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{H})$.