  We solve the spectrum of quantum spin chains based on representations of the
Temperley-Lieb algebra associated with the quantum groups ${\cal U}%
_{q}(X_{n})$ for $X_{n}=A_{1},$ $B_{n},$ $C_{n}$ and $D_{n}$. The tool is a
modified version of the coordinate Bethe Ansatz through a suitable choice of
the Bethe states which give to all models the same status relative to their
diagonalization. All these models have equivalent spectra up to degeneracies
and the spectra of the lower dimensional representations are contained in the
higher-dimensional ones. Periodic boundary conditions, free boundary conditions
and closed non-local boundary conditions are considered. Periodic boundary
conditions, unlike free boundary conditions, break quantum group invariance.
For closed non-local cases the models are quantum group invariant as well as
periodic in a certain sense.
