The algebra $U_q({\mathfrak{s}\mathfrak{l}}_2)$ in disguise

We discuss a connection between the algebra $U_q({\mathfrak{s}\mathfrak{l}}_2)$ and the tridiagonal pairs of $q$-Racah type. To describe the connection, let $x,y^{\pm 1},z$denotetheequitablegeneratorsfor$U_q({\mathfrak{sl}_2})$\mathrm{.}Let$U^\vee_q$denotethesubalgebraof$U_q({\mathfrak{sl}_2})$generatedby$x,y^{-1},z$\mathrm{.}Usingatridiagonalpairof$q$-Racah type we construct two finite-dimensional $U_q^{\vee }$-modules. The constructions yield two nonstandard presentations of $U_q^{\vee }$ by generators and relations. These presentations are investigated in detail.