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We interpret the Rahman polynomials in terms of the Lie algebra $sl_3(C)$.Using the parameters of the polynomials we define two Cartan subalgebras for$sl_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism$\dagger$ of $sl_3(C)$ that fixes each element of $H$ and each element of$\tilde{H}$. We consider a certain finite-dimensional irreducible$sl_3(C)$-module $V$ consisting of homogeneous polynomials in three variables.We display a nondegenerate symmetric bilinear form $<,>$ on $V$ such that$<\beta \xi,\zeta> = < \xi,\beta^\dagger \zeta>$ for all $\beta \in sl_3(C)$and $\xi,\zeta \in V$. We display two bases for $V$; one diagonalizes $H$ andthe other diagonalizes $\tilde{H}$. Both bases are orthogonal with respect to$<,>$. We show that when $<,>$ is applied to a vector in each basis, the resultis a trivial factor times a Rahman polynomial evaluated at an appropriateargument. Thus for both transition matrices between the bases each entry isdescribed by a Rahman polynomial. From these results we recover the previouslyknown orthogonality relation for the Rahman polynomials. We also obtain twoseven-term recurrence relations satisfied by the Rahman polynomials, along withthe corresponding relations satisfied by the dual polynomials. These recurrencerelations show that the Rahman polynomials are bispectral. In our theory theroles of $H$ and $\tilde{H}$ are interchangable, and for us this explains theduality and bispectrality of the Rahman polynomials. We view the action of $H$and $\tilde{H}$ on $V$ as a rank 2 generalization of a Leonard pair.

The Rahman polynomials and the Lie algebra sl_3(C)