Short Hand Notation and Rules
Poset of Abelian Radical
Geometrically, Abelian Radical is the face of the Root polyhedron which contains (at long end of Dynkin graph) and is parallel to the hyperplane spanned by the remaining roots. Algebraically, it consists of roots which -coefficient is 1. Explicitly
Size of the radical:
The Levi factor of the parabolic subalgebra defined by is . This gives a 27-dimensional representation of (its smallest fundamental one).