Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach in characterizing Lie polynomials the generators $A,B$ of algebra with a defining relation that is form deformed or twisted commutation $AB=\sigma(BA)$ where deformation twisting map $\sigma$ linear polynomial slope parameter not root unity. The class algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation and some types $q$-oscillator whose parameters are roots unity, so we have general solution for characterization problem these algebras.