Show that in any finite field, there exists a non-trivial solution for $x^2 + y^2 + z^2 + w^2 = 0$.
I have shown it for finite fields of cardinality $q$, when $4$ divides $q-1$ and when $q-1$ is odd. In such finite fields, $-1$ has a square root. Hence, $x^2+y^2 = 0$ has a non-trivial solution.
I want a non-trivial solution for the above equation when $q-1 \equiv 2 (\mod 4)$? Here, we already know that there is no non-trivial solution for $x^2+y^2 = 0$.
Is there a more general approach that does not care about the cardinality of the field?