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We give a classification of the matrices in the unitary group U(1,1;H),whereH is the division ring of the real quaternions. To this end, we consider thecomplex representation phi(P) for P in U(1,1;H). Next, we compute thecharacteristic polynomial f of the 4x4 complex matrix phi(P) and then study thesign of the resultant of f and its derivative f'.

Isometries of the quaternionic hyperbolic line