Algebraic characterization of reversibility in the quaternionic M\"obius group

An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic M\"{o}bius group $\mathrm{PSL}(2,\mathbb{H})$ 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 M\"{o}bius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that $$[A]\in \mathrm{PSL}(2,\mathbb{H}) \text{ is reversible} \quad \Longleftrightarrow \quad \beta_A^{2}=\delta_A^{2},$$ where $\beta_A$ and $\delta_A$ are real conjugacy invariants associated with a lift $A\in \mathrm{SL}(2,\mathbb{H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in $\mathrm{SL}(2,\mathbb{H})$ and $\mathrm{PSL}(2,\mathbb{H})$.