Algebraic characterization of the isometries of the hyperbolic 5-space

Let GL(2, H) be the group of invertible 2 x 2 matrices over the division algebra H of quaternions. GL(2, H) acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in GL(2, H).