A Laurent Phenomenon for the Cayley Plane

We describe a Laurent phenomenon for the Cayley plane, which is the homogeneous variety associated to the cominuscule representation of $E_6$. The corresponding Laurent phenomenon algebra has finite type and appears in a natural sequence of LPAs indexed by the $E_n$ Dynkin diagrams for $n\le 6$. We conjecture the existence of a further finite type LPA, associated to the Freudenthal variety of type $E_7$.