  An algebra is introduced which can be considered as a rank 2 extension of the
Askey-Wilson algebra. Relations in this algebra are motivated by relations
between coproducts of twisted primitive elements in the two-fold tensor product
of the quantum algebra $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb C))$. It is
shown that bivariate $q$-Racah polynomials appear as overlap coefficients of
eigenvectors of generators of the algebra. Furthermore, the corresponding
$q$-difference operators are calculated using the defining relations of the
algebra, showing that it encodes the bispectral properties of the bivariate
$q$-Racah polynomials.
