The product of the sums of two squares is…

This may not come as a great surprise to anyone else, but I was mildly pleased to stumble on the fact (?) that if two integers are both the sums of two squares, so is their product:

If

x = a² + b²

and 

y = c² + d²

then 

xy =

(a² + b²)(c² +d²) =

a²(c² + d²) + b²(c² +d²) =

a²c² + a²d² + b²c² + b²d² =

(a²c² + b²d²⁾ + (a²d² + b²c²⁾ =

(a²c² +²abcd + b²d²) + (a²d² -²abcd + b²c²) =

(ac + bd)² + (ad - bc)²

and by a similar procedure we can show that
xy = (ac - bd)² + (ad + bc)²

Thus xy is actually the sum of two pairs of squares, unless ad = bc or ac = bd.

Or so it seems to me.