Conditions on x,y, and z in the Lipschitz quaternions where x2+y2=z2

The integral solutions to x²+y²=z² are called Pythagorean triples, and the solutions are parameterized by x=a(m²-n²), y=2amn, z=a(m²+n²) for integers a,m,n, where m>n>0, m, n relatively prime, and exactly one of m,n is even. This result has been extended to the Gausssian integers as well as to unique factorization domains.
In this work, we explore solutions to the equation x²+y²=z² in the ring of Lipschitz (integral) quaternions. Specifically, we have found necessary conditions for x,y and z in the cases that x and y are odd, and z is even; that exactly one of x and y is even, and z is odd; or that all three of x,y and z are even. These conditions help in the search for a parameterization of all Pythagorean triples over Lipshitz quaternions.