  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 $H$ and
the other diagonalizes $\tilde{H}$. Both bases are orthogonal with respect to
$<,>$. We show that when $<,>$ is applied to a vector in each basis, the result
is a trivial factor times a Rahman polynomial evaluated at an appropriate
argument. Thus for both transition matrices between the bases each entry is
described by a Rahman polynomial. From these results we recover the previously
known orthogonality relation for the Rahman polynomials. We also obtain two
seven-term recurrence relations satisfied by the Rahman polynomials, along with
the corresponding relations satisfied by the dual polynomials. These recurrence
relations show that the Rahman polynomials are bispectral. In our theory the
roles of $H$ and $\tilde{H}$ are interchangable, and for us this explains the
duality 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.
