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Motivated by the recent advances in the categorification of the clusterstructure on the coordinate rings of Grassmannians of $k$-subspaces in$n$-space, we investigate a particular construction of root systems of type$\mathsf{T}_{2,p,q}$, including the type $\mathsf{E}_n$. This constructiongeneralizes Manin's ``hyperbolic construction'' of $\mathsf{E}_8$ and reveals alot of otherwise hidden regularities in this family of root systems.

Real roots in the root system $\mathsf{T}_{2,p,q}$