  The notion of factorized $A_2$-Leonard pair is introduced. It is defined as a
rank 2 Leonard pair, with actions in certain bases corresponding to the root
system of the Weyl group $A_2$, and with some additional properties. The
functions arising as entries of transition matrices are bivariate orthogonal
polynomials (of Tratnik type) with bispectral properties. Examples of
factorized $A_2$-Leonard pairs are constructed using classical Leonard pairs
associated to families of orthogonal polynomials of the ($q$-)Askey scheme. The
most general examples are associated to an intricate product of univariate
($q$-)Hahn and dual ($q$-)Hahn polynomials.
