  Fock and Goncharov introduced cluster ensembles, providing a framework for
coordinates on varieties of surface representations into Lie groups, as well as
a complete construction for groups of type $A_n$. Later, Zickert, Le, and Ip
described, using differing methods, how to apply this framework for other Lie
group types. Zickert also showed that this framework applies to triangulated
$3$-manifolds. We present a complete, general construction, based on work of
Fomin and Zelevinsky. In particular, we complete the picture for the remaining
cases: Lie groups of types $F_4$, $E_6$, $E_7$, and $E_8$.
