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

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