Lie algebras with Abelian centralizers

In the paper, finite-dimensional real Lie algebras for which the centralizers of all nonzero element are Abelian are studied. These Lie algebras are also characterized by the transitivity condition for the commutation relation for two nonzero elements. A complete description of these Lie algebras up to isomorphism is given. Some results concerning the relationship between the aforementioned Lie algebras and the Lie algebras of vector fields whose orbits are one-dimensional are considered.