  We discuss a connection between the algebra $U_q({\mathfrak{sl}_2})$ and the
tridiagonal pairs of $q$-Racah type. To describe the connection, let $x,y^{\pm
1},z$ denote the equitable generators for $U_q({\mathfrak{sl}_2})$. Let
$U^\vee_q$ denote the subalgebra of $U_q({\mathfrak{sl}_2})$ generated by
$x,y^{-1},z$. Using a tridiagonal pair of $q$-Racah type we construct two
finite-dimensional $U^\vee_q$-modules. The constructions yield two nonstandard
presentations of $U^\vee_q$ by generators and relations. These presentations
are investigated in detail.
