Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials $$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$ I'm interessted in $\xi_0,\xi_1,\xi_2$ and relations between them. Using a computer algebra system I checked that $\xi_2$ is in $(\xi_1,\xi_0)$, the ideal generated by $\xi_0,\xi_1$, for $3 \leq q \leq 37$. Therefore I have the impression that $\xi_2 \in (\xi_0,\xi_1)$ for all $q$. Unfortunately I was not able to prove it.

What I have tried: For $3 \leq q \leq 37$ I (my computer) computed polynomials $f,g \in \mathbb F_q[x_1,x_2,x_3,x_4]$ satisfying $\xi_2 = f \xi_1 + g \xi_0$. Since they are too complicated I can not recognize any pattern or general form for arbitrary $q$.

Motivation: The polynomials $\xi_0,\xi_1,\xi_2$ are part of a generating system of the polynomial invariants of the four-dimensional orthogonal group of plus type:

Chu , H., 2001, Polynomial invariants of four-dimensional orthogonal groups. Comm. Algebra 29, 1153–1164, http://dx.doi.org/10.1081/AGB-100001673

Long story short: Does anyone know how to show $$x_1^{q^2+1} - x_2^{q^2+1} + x_3^{q^2+1} - x_4^{q^2+1} \in ( x_1^{2} - x_2^{2} + x_3^{2} - x_4^{2}, x_1^{q+1} - x_2^{q+1} + x_3^{q+1} - x_4^{q+1})$$ over a finite field with $q$ elements? Or maybe it is just wrong for arbitrary $q$? Thank you in advance.