  We here construct an explicit isomorphism between any commutative Hopf
algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and
the shuffle algebra based on the same space. This isomorphism uses the
commutative $B_\infty$ structure that governs the product and the eulerian
idempotent, as well as the canonical projection on the space $V$. This
generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle
algebras, which correspond to the case when the $B_\infty$ structure is given
by an associative and commutative product. We develop several examples in
details, including the Hopf algebra of finite topologies.
