Uniqueness of addition in Lie algebras revisited

We obtain new and improve old results on the uniqueness of addition in Lie rings and Lie algebras. A Lie ring ℜ is called a unique addition Lie ring, or a UA-Lie ring, if any commutator-preserving bijection from ℜ to an arbitrary Lie ring is additive. We describe wide classes of Lie rings that are not UA-Lie rings. On the other hand, it is known that if a finite-dimensional Lie algebra ???? contains two elements whose centralizers have trivial intersection, then ???? is a UA-Lie ring. We use this result to characterize UA-Lie rings among seaweed Lie algebras. The paper includes many open problems and questions.