The Lie algebra associated to a unit form

To any unit form , , we associate a Lie algebra —an intersection matrix Lie algebra in the terminology of Slodowy—by means of generalized Serre relations. For a nonnegative unit form the isomorphism type of is determined by the equivalence class of q. Moreover for q nonnegative and connected with radical of rank zero or one respectively, the algebras turn out to be exactly the simply-laced Lie algebras which are finite-dimensional simple or affine Kac–Moody, respectively. In case q is connected, nonnegative of corank two and not of Dynkin type , the algebra G(q) is elliptic.