Write $(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$ as a sum of (three) squares of quadratic forms The quartic form 
$$(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$$ 
is non-negative for all real $x$, $y$, $z$, as one can check (with some effort). A theorem of Hilbert implies that there exist quadratic forms $Q_1$, $Q_2$, $Q_3$ so that
$$(x^2 + y^2 + z^2)^2 - 3( x^3 y + y^3 z + z^3 x) = Q_1^2 + Q_2^2 + Q_3^2$$
I would like to find an explicit writing of the quartic forms, with rational quadratic forms $Q_i$. Maybe more than $3$ terms are necessary.  

$$(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)=\frac{1}{2}\sum_{cyc}(x^2-y^2-xy-xz+2yz)^2=$$
$$=\frac{1}{6}\sum_{cyc}(x^2-2y^2+z^2-3xz+3yz)^2.$$
