  Let $q$ be a nonzero complex number that is not a root of unity. In the
$q$-oscillator with commutation relation $ a a^+-qa^+ 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 ${a^+}^l{\left[a,a^+\right]}^k$, and
${\left[a,a^+\right]}^k a ^l$, 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 $h\geq 2$ 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 $AW(3)$.
