  The higher rank Askey-Wilson algebra was recently constructed in the $n$-fold
tensor product of $U_q(\mathfrak{sl}_2)$. In this paper we prove a class of
identities inside this algebra, which generalize the defining relations of the
rank one Askey-Wilson algebra. We extend the known construction algorithm by
several equivalent methods, using a novel coaction. These allow to simplify
calculations significantly. At the same time, this provides a proof of the
corresponding relations for the higher rank $q$-Bannai-Ito algebra.
