Response to query

I asked a friend to search for solutions between 1 and 1000 to $a^2+b^2 = c^2+d^2 = e^2+f^2 = g^2+h^2$ and $a^4+b^4+c^4+d^4 = e^4+f^4+g^4+h^4$ he found this one $106\ 237\ 189\ 178\ 141\ 218\ 126\ 227$, this means you can have 4 quadratics with 16 integer roots $\begin{gather} (-106, 106), (-237, 237), (-189, 189), (-178, 178), (-141, 141), (-218, 218), (-126, 126), (-227, 227), \\ (-11236, -56169), (-35721, -31684), (-19881, -47524), (-15876, -51529), \\ (631114884, 1131784164), (944824644, 818074404). \end{gather}$

PD: second row are the roots of the composition of first three quadratics, third row are the roots of composition of first two quadratics.

PD2: each row is obtained from the preceding one by multiplying consecutive terms. note that for each row, the sum of terms between brackets is constant

Carlos di Fiore