Multiplicative functions which are additive on sums of two nonzero squares

Let $f$ be a multiplicative function which satisfies $$f(a^2+b^2+c^2+d^2)=f(a^2+b^2)+f(c^2+d^2)$$ for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)f(11)\mathrm{\ne }0$. Otherwise, $f(n)=0$ for all $n\ge 2$ except for $n=3,9,11$.