Classification and Decomposition of Quaternionic Projective Transformations

We consider the projective linear group $\mathrm{P}\mathrm{S}\mathrm{L}(3,\mathbb{H})$. We have investigated the reversibility problem in this group and use the reversibility to offer an algebraic characterization of the dynamical types of $\mathrm{P}\mathrm{S}\mathrm{L}(3,\mathbb{H})$. We further decompose elements of $\mathrm{S}\mathrm{L}(3,\mathbb{H})$ as products of simple elements, where an element $g$ in $\mathrm{S}\mathrm{L}(3,\mathbb{H})$ is called $\mathit{\text{simple}}$ if it is conjugate to an element of $\mathrm{S}\mathrm{L}(3,\mathbb{R})$. We have also revisited real projective transformations and following Goldman’s ideas, have offered a complete classification for elements of $\mathrm{S}\mathrm{L}(3,\mathbb{R})$.