TITLE: Express $x^4 + y^4 + x^2 + y^2$ as sum of squares of three polynomials in $x,y$
QUESTION [10 upvotes]: I don't know any identity that'd help me simplify it. I know of Brahmagupta's identity and tried using but no good. Any hints? 
Edit: So far I've tried various things, 
$(x^2+1/2)^2 + (y^2 +1/2)^2 -1/2$
This doesn't get me anywhere.
I tried solving in x as a quadratic so as to get an insight or something but that didn't help too.
I even tried writing general polynomials of degree too and then getting the solution by comparing coefficients, but that wasn't elegant and was too computational.

REPLY [6 votes]: let real constant
$$  t = \sqrt{\sqrt 8 - 2} $$
so that
$$  \frac{t^4}{4} + t^2 = 1. $$
Then
$$ \left(-x^2 + \frac{t^2}{2} y^2 \right)^2 +  \left( t y^2 + x \right)^2 +  \left( txy - y \right)^2 = x^4 + y^4 + x^2 + y^2 $$
ADDED: apparently this was asked and answered six years ago, and I commented  there.