(untitled) (Communications in Mathematical Research, 2021)
In this paper, through a meticulous description of finite root system, a concrete comultiplication with an explicit action on the basis elements of finite dimensional simple Lie algebras of type A, D, E is constructed. Then any finite dimensional simple Lie algebra of type A, D, E is endowed with a new generalized Lie coalgebra splitting. This construction verifies the known existence of a co-split Lie structure on any finite dimensional complex simple Lie algebra.