  We generalize to the non simply-laced case results of Gei\ss, Leclerc and
Schr\"oer about the cluster structure of the coordinate ring of the maximal
unipotent subgroups of simple Lie groups. In this way, cluster structures in
the non simply-laced case can be seen as projections of cluster structures in
the simply-laced case. This allows us to prove that cluster monomials are
linearly independent in the non simply-laced case.
