In the [webpage](https://mathoverflow.net/questions/131371/non-crystallographic-cluster-algebras), there is a result:


Theorem 1. Coefficient free cluster algebras without frozen variables are in bijection with Dynkin diagrams of type $A\_n$, $B\_n$, $C\_n$, $D\_n$, $E\_6, E\_7, E\_8$, $F\_4$, $G\_2$. 


On the other hand, on page 46 of the [lecture notes](https://www.msri.org/workshops/595/schedules/15561/documents/1438/assets/16934), there is a result:


Theorem 2. A cluster algebra of finite type ifand only if the mutable part of its quiver at some seed is an orientation of a simply-laced Dynkin diagram.


My question is: cluster algebras of finite type corresponds to Dynkin diagrams or only simply-laced Dynkin diagrams? Thank you very much.



Cluster algebras of finite type
You have a finite type cluster algebra associated to every Cartan matrix, regardless whether the Dynkin diagram is simply-laced or not.


The cluster algebras of types $B\_n$ and $C\_n$ have been studied for example in <http://www.dmtcs.org/pdfpapers/dmAJ0138.pdf> It is known that the cluster algebra $B\_2$ is the coordinate ring of $Sp\_4/N$.


