  Let $\mathbb{F}$ be a field, and fix a $q\in\mathbb{F}$. The $q$-deformed
Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra over
$\mathbb{F}$ with generators $A$, $B$ and a relation which asserts that $AB -
qBA$ is the multiplicative identity in $\mathcal{H}(q)$. We extend
$\mathcal{H}(q)$ into an algebra $\mathcal{R}(q)$ defined by generators $A$,
$B$ and a relation which asserts that $AB-qBA$ is central in $\mathcal{R}(q)$.
We identify all elements of $\mathcal{R}(q)$ that are Lie polynomials in $A$,
$B$.
