  We show how the multiplicative structure on Pythagorean triples defined by
Beauregard--Suryanarayan arises from automorphisms of the hyperbolic plane, and
how the structure theorem of this product follow naturally from this
connection. We also show how the natural involution on Pythagorean triples is
related to Pell's equation with the parameter 2, and prove a generalization of
this phenomenon.
