  The equitable presentation of $U_q(\mathfrak{sl}_2)$ was introduced in 2006
by Ito, Terwilliger, and Weng. This presentation involves some generators $x,
y, y^{-1}, z$. It is known that $\{x^r y^s z^t : r, t \in \mathbb{N}, s \in
\mathbb{Z}\}$ is a basis for the $\mathbb{K}$-vector space
$U_q(\mathfrak{sl}_2)$. In 2013, Bockting-Conrad and Terwilliger introduced a
subalgebra $\mathcal{A}$ of $U_q(\mathfrak{sl}_2)$ spanned by the elements
$\{x^r y^s z^t : r, s, t \in \mathbb{N}, r+s+t \ {\rm even}\}$. We give a
presentation of $\mathcal{A}$ by generators and relations. We also classify up
to isomorphism the finite-dimensional irreducible $\mathcal{A}$-modules, under
the assumption that $q$ is not a root of unity.
