Parameterization of Pythagorean n-tuples

Finding parameterizations for Pythagorean triples for the equation,  x²  + y² = z², is a well-known problem in number theory. Recently, research has been conducted concerning parameterization techniques for up to six variables. An interesting article on this topic was published in 2010 by Sophie Frisch and Leonid Vaserstein which provided guidelines to parameterize Pythagorean quadruples, quintuples and sextuples [reference: Polynomial parameterization of Pythagorean quadruples, quintuples and sextuples]. The method of producing sextuples involved the equation , x₁² + .....+  x_(n-1)²= x_n², solutions to which are represented by the determinant of a hermitian matrix , the form of which captures essential information about the equation. As a result of the work of Frisch and Vaserstein, an interesting problem arose, which namely is the question of whether or not a similar method can be used to produce similarly representative hermitian matrices for equations in  variables. Important to this process is the fact that, for instance,  matrices with Gaussian integer entries can be represented as  matrices with real entries. In answering this question, I came up with partial results showing that it is possible to produce Pythagorean n-tuples as long as certain conditions are imposed on the hermitian matrix. In future, I would like to extend my research in generalizing the partial results that I already came up with.