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Denote by $\mathbb{H}$ the set of all quaternions. We are interested in thegroup $U(1,1;\mathbb{H})$, which is a subgroup of $2\times 2$ quaternionicmatrix group and is sometimes called $Sp(1,1)$. As well known,$U(1,1;\mathbb{H})$ corresponds to the quaternionic M\"{o}bius transformationson 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 $T\inU(1,1;\mathbb{H})$, which corresponds to an elliptic quaternionic M\"{o}biustransformation $g_T(z)$, could be $U(1,1;\mathbb{H})$-similar to a diagonalmatrix.

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