  In this paper, we review a few known facts on the coordinate Bethe ansatz. We
present a detailed construction of the Bethe ansatz vector $\psi$ and energy
$\Lambda$, which satisfy $V \psi = \Lambda \psi$, where $V$ is the the transfer
matrix of the six-vertex model on a finite square lattice with periodic
boundary conditions for weights $a= b=1$ and $c > 0$. We also show that the
same vector $\psi$ satisfies $H \psi = E \psi$, where $H$ is the Hamiltonian of
the XXZ model (which is the model for which the Bethe ansatz was first
developed), with a value $E$ computed explicitly.
  Variants of this approach have become central techniques for the study of
exactly solvable statistical mechanics models in both the physics and
mathematics communities. Our aim in this paper is to provide a
pedagogically-minded exposition of this construction, aimed at a mathematical
audience. It also provides the opportunity to introduce the notation and
framework which will be used in a subsequent paper by the authors that amounts
to proving that the random cluster model on $\mathbb{Z}^2$ with cluster weight
$q >4$ exhibits a first-order phase transition.
