Pythagorean Triples and the Square Root of a Pure Imaginary Integer Quaternion

Solutions to the n=2 case of Fermat's equation, x^n+y^n=z^n, in several commutative rings are completely parameterized-such is the case for the integers, Gaussian integers, and more generally, in any unique factorization domain. Over non-commutative rings however, this terrain remains vastly unexplored. A Pythagorean triple in an arbitrary ring is defined as in the integer case. We aim to find a family of Pythagorean triples over the ring of Lipschitz (integer) Quaternions. To this end, we describe a class of pure imaginary integer Quaternion Pythagorean triples. We also show that if q=z^2 for a pure imaginary integer Quaternion q and integer Quaternion z, then the coefficients of z and q must satisfy a system of four quadratic Diophantine equations in four indeterminants. Our present work centers around finding sufficient conditions on the coefficients of q so that z=sqrt{q} be an integer Quaternion.