A ring of Pythagorean triples over quadratic fields (Q2259510)

From the text: A triple \(\langle \alpha,\beta,\gamma\rangle\) of elements of a ring is called a Pythagorean triple if \(\alpha^2 + \beta^2 = \gamma^2\). \textit{B. Dawson} [Missouri J. Math. Sci. 6, No. 2, 72--77 (1994; Zbl 1097.11527)] defined operations on the set of all Pythagorean triples in \(\mathbb Z\) so that this set is a ring. \textit{J. T. Cross} [Math. Mag. 59, 106--110 (1986; Zbl 0601.10010)] displayed a method for generating all Pythagorean triples over the ring of Gaussian integers. Let \(K\) be a quadratic field and let \(R\) be the ring of integers of \(K\) such that \(R\) is a unique factorization domain. The set \(P\) of all Pythagorean triples in \(R\) is partitioned into \(P_\eta\), sets of triples \(\langle\alpha,\beta,\gamma\rangle\) in \(P\) where \(\eta=\gamma-\beta\). We show the ring structures of each \(P_\eta\) and \(P\) from the ring structure of \(R\).