Families of vector fields which generate the group of diffeomorphisms

Given a compact manifold $M$ and a family of vector fields $\mathcal F$ such that the group generated by $\mathcal F$ acts transitively on $M$, we prove that the group of all diffeomorphisms of $M$ that are isotopic to the identity is generated by the exponentials of vector fields in $\mathcal F$ rescaled by smooth functions.