I wonder if someone could shed some light in the following question
Let $(x,y)$ denote the greatest common divisor of $x$ and $y$, and let $x_1,y_1,x_2,y_2$ be integers.
Is the following statement true?
If $(x_1,y_1)=(x_2,y_2)=(x_1^2+y_1^2,x_2^2+y_2^2)=1$, then $$(x_1x_2\pm y_1y_2, x_1y_2\mp x_2y_1)=1$$
If not, what further hypotheses are necessary to guarantee the claim?
Thanks in advance, Guillermo