A polynomial analogue of berggrens theorem on pythagorean triples

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x(2) + y(2) = z(2). An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3,4,5) and (4,3,5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren ' s theorem in the context of a one-variable polynomial ring over a field of characteristic not equal 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.