Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $a{a}^{+}-q{a}^{+}a=1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^{+}$ and $a$ is the linear span of $a^{+}$ and $a$, together with all operators of the form ${a^{+}}^l{[a,a^{+}]}^k$, and ${[a,a^{+}]}^k{a}^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $a^h$ or $(a^{+}{)}^h$ with $h\ge 2$ or $h=0$ are outside the commutator algebra generated by $a$ and $a^{+}$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^{+}$ and $a$. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $a$, $a^{+}$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.