Although the open XXZ spin chain with non-diagonal boundary terms is known to be integrable, the corresponding Bethe ansatz equations cannot be derived directly because of the lack of a suitable reference state. They also present many unusual features, among which an “inhomogeneous” term which vanishes only if the parameters of the model satisfy a certain quantization condition known as the Nepomechie constraint. Reinterpreting the Hamiltonian of this spin chain as an abstract element of the two-boundary Temperley-Lieb algebra evaluated in its