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 Möbius transformations on the unit ball in \(\mathbb {H}\). In this article, some similarity 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 Möbius transformation \(g_T(z)\), could be \(U(1,1;\mathbb {H})\)-similar to a diagonal matrix. Moreover, one can see that each elliptic quaternionic Möbius transformation is quaternionic Möbius conjugate to a bi-rotation, where a bi-rotation means a map \(z\rightarrow p\cdot z \cdot q^{-1}\) for some \(p,q\in \mathbb {H}\) with \(|p|=|q|=1\).