  In this paper we give the global conditions for an ordinary differential
equation to admit a superposition law of solutions in the classical sense. This
completes the well-known Lie superposition theorem. We introduce rigorous
notions of pretransitive Lie group action and Lie-Vessiot systems. We proof
that pretransitive Lie group actions are transitive. We proof that an ordinary
differential equation admit a superposition law if and only if it is a
pretransitive Lie-Vessiot system. It means that its enveloping algebra is
spanned by fundamental fields of a pretransitive Lie group action. We discuss
the relationship of superposition laws with differential Galois theory and
review the classical result of Lie.
