  The paper is devoted to prove a version of Milnor-Moore Theorem for connected
braided bialgebras that are infinitesimally cocommutative. Namely in
characteristic different from 2, we prove that, for a given connected braided
bialgebra $A$ having a $\lambda $-cocommutative infinitesimal braiding for some
regular element $\lambda \neq 0$ in the base field, then the infinitesimal
braiding of $A$ is of Hecke-type of mark $\lambda$ and $A$ is isomorphic as a
braided bialgebra to the symmetric algebra of the braided subspace of its
primitive elements.
