With the basic set for doing a product of sum of squares you can build bigger ones easily.
And went for LOTS of easy with this example as could make something that looks harder but is extra work for nothing. That would be an illusion as is so easy to do.
(x2 + 2y2)(u2 + 3v2)(x'2 + 4y'2)(u'2 + 5v'2) = p2 + 359q2
And finding integer solutions for all the variables is easy. Coming up with variables is harder. But yeah you can just keep going as far as you want.
x = 1, y = 2, u = 2, v = 2, x' = 3, y' = 2, u' = 4, v' = 2, p = 358, q = 2
Using first iteration and using positive as all will get squared. So:
(1 + 8)(4 + 12)(9 + 16)(16 + 20) = (3582 + 359*4)
(9)(16)(25)(36) = (3582 + 359*4) = 129600
Which worked. Used easy first iterations. Of course iterators work out to infinity.
Just playing around.