Algebraic characterization of reversibility in the quaternionic Möbius group

An element of a group is called reversible if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group PSL(2,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 \[ [A]∈ PSL(2,H) is reversible β_A²=δ_A², \] where βA and δA are real conjugacy invariants associated with a lift A∈ SL(2,H). Furthermore, we give a complete characterization of reversing symmetries of reversible elements in SL(2,H) and PSL(2,H).