Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples

For n=4 or 6, the Pythagorean n-tuples admit a parametrization by a single n-tuple of polynomials with integer coefficients (which is impossible for n=3). For n=5, there is an integer-valued polynomial Pythagorean 5-tuple which parametrizes Pythagorean 5-tuples (similar to the case n=3). Pythagorean quadruples are closely related to (integer) Descartes quadruples, which we also parametrize by a Descartes quadruple of polynomials with integer coefficients.