  The Askey--Wilson algebras were used to interpret the algebraic structure
hidden in the Racah--Wigner coefficients of the quantum algebra
$U_q(\mathfrak{sl}_2)$. In this paper, we display an injection of a universal
analog $\triangle_q$ of Askey--Wilson algebras into
$U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$
behind the application. Moreover we formulate the decomposition rules for
$3$-fold tensor products of irreducible Verma $U_q(\mathfrak{sl}_2)$-modules
and of finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules into the
direct sums of finite-dimensional irreducible $\triangle_q$-modules.
