Some invariants of $U(1,1;\mathbb{H})$ and diagonalization

Denote by $\mathbb{H}$ the set of all quaternions. We are interested in the group $U(1,1;\mathbb{H})$, which is a subgroup of $2\times 2$ quaternionic matrix group and is sometimes called $Sp(1,1)$. As well known, $U(1,1;\mathbb{H})$ corresponds to the quaternionic M\"{o}bius transformations on the unit ball in $\mathbb{H}$. In this article, some similar invariants on $U(1,1;\mathbb{H})$ are discussed. Our main result shows that each matrix $T\in U(1,1;\mathbb{H})$, which corresponds to an elliptic quaternionic M\"{o}bius transformation $g_T(z)$, could be $U(1,1;\mathbb{H})$-similar to a diagonal matrix.