Note on the identity function

Abstract. We consider the functional equation $$ f\left(n^2+m^2+a+b\right)=g(n^2+a)+g(m^2+b)\quad\hbox{for all}\quad n,m\in{\Bbb N}, $$ where \(a,b\) are non-negative integers with \(a+b >0\) and \(f,g\) are multiplicative functions.