I'd like to know if the only 'sum of two squares' expressions for $(a^2+b^2)(c^2+d^2)$ are $(ac\pm bd)^2 + (ad\mp bc)^2$. (Expression that works generally for all integers $a, b, c, d$) It looks pretty true to me since numbers like $(2^2+1^2)(3^2+1^2)=50$ can only be written in the sum of two squares as $(2\times3+1\times1)^2+(2\times1-1\times3)^2=7^2+1^2$ or $(2\times3-1\times1)^2+(2\times1+1\times3)^2=5^2+5^2$
Any help would be greatly appreciated.