Common left- and right-hand divisors of a quaternion integer

Given a quaternion integer α whose norm is divisible by a natural number m, does there exist a quaternion integer β of norm m dividing α on both the left and right? This problem is a case of the “metacommutation problem”, which asks generally for relationships between the many different factorizations of a given integral quaternion. In this paper, we give necessary and sufficient conditions on primitive α of odd norm to ensure the existence of common left- and right-hand divisors, and we characterize the non-trivial sets of such divisors.