A polynomial analogue of Berggren's 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 $\mathrm{\ne }2$. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.