Pythagorean Triples and Units in Integral Group Rings

In this note, we describe a simple method for finding units of group rings of the form Z[G] = Z[H]#Z[C2] for H an abelian group, and apply this to the case G = D4, the dihedral group of order 8. Here, units may be described as integer points on hyperboloids, and, defining units u,v to be equivalent if they differ by an inner automorphism, we see that this equivalence relation partitions each hyperboloid into finitely many classes.