Lie polynomials in a $q$-deformed universal enveloping algebra of a low-dimensional Lie algebra

The nonabelian two-dimensional Lie algebra over a field $\mathbb{F}$ has a presentation by generators $A$, $B$ and relation $[A,B]=A$, with the universal enveloping algebra having a presentation by generators $A$, $B$ and relation $AB-BA=A$. A well-known fact is that the said Lie algebra is isomorphic to that which has a universal enveloping algebra that has a presentation by generators $A$, $B$ and relation $AB-BA=B$. Given $q,r,s\in \mathbb{F}$, solutions to the Lie polynomial characterization problems in the corresponding $q$-deformed universal enveloping algebras, with generalized relations $AB-qBA=rA$ and $AB-qBA=sB$, respectively, are presented.