  We obtain new and improve old results on uniqueness of addition in Lie rings
and Lie algebras. A Lie ring $\mathfrak{R}$ is called a unique addition ring,
or a UA-Lie ring, if any commutator-preserving bijection from $\mathfrak{R}$ to
an arbitrary Lie ring is additive. We describe wide classes of Lie rings that
are not UA-Lie ring. In the other direction, it is known that if a
finite-dimensional Lie algebra $\mathfrak{g}$ contains two elements whose
centralizers have trivial intersection, then $\mathfrak{g}$ 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.
