Multiplicative functions commutable with binary quadratic forms $x^2\pm xy+y^2$

If a multiplicative function $f$ is commutable with a quadratic form $x^2+xy+y^2$, i.e., $$f(x^2+xy+y^2)=f(x{)}^2+f(x)f(y)+f(y{)}^2,$$ then $f$ is the identity function. In other hand, if $f$ is commutable with a quadratic form $x^2-xy+y^2$, then $f$ is one of three kinds of functions: the identity function, the constant function, and an indicator function for $\mathbb{N}\setminus p\mathbb{N}$ with a prime $p$.