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