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

The Bethe ansatz for the six-vertex and XXZ models: an exposition