Equivalence of Vector Field Realizations of Lie Algebras from the Lie Group Point of View

The equivalence of vector field realizations of Lie algebras is considered from the viewpoint of the Lie algebra and also from the corresponding (local) Lie group. It is shown that for some stages in the establishment of the equivalence things are simpler in the case of the Lie group. A lemma about the equivalence of the sum of realizations is proposed.