Splints of root systems for special Lie subalgebras

Splint is a decomposition of root system into union of root systems. Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We consider special embedding of Lie subalgebra to Lie algebra. We classify projections of algebra root systems using extended Dynkin diagrams and single out the conditions of splint appearance and coincidence of branching coefficients with weight multiplicities. While such a coincidence is not very common it is connected with Gelfand-Tsetlin basis.