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The Askey--Wilson algebras were used to interpret the algebraic structurehidden in the Racah--Wigner coefficients of the quantum algebra$U_q(\mathfrak{sl}_2)$. In this paper, we display an injection of a universalanalog $\triangle_q$ 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 $U_q(\mathfrak{sl}_2)$-modulesand of finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules into thedirect sums of finite-dimensional irreducible $\triangle_q$-modules.

An embedding of the universal Askey-Wilson algebra into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$