Theorem 1

Let E be a cocommutative bigebra. (a) The bigebra morphism $f_E : U(P(E)) \to E$ is injective. (b) If there exists on E a filtration compatible with its bigebra structure (no. 3, Definition 2), the morphism $f_E$ is an isomorphism. (In case (b), the bigebra E is therefore identified with the enveloping bigebra of the Lie algebra of its primitive elements.) Let $c_E$ (resp. $\varepsilon_E$) be the coproduct (resp. counit) of E. We write $g = P(E)$; let $(e_i)_{i \in I}$ be a basis of the vector K-space g, where the indexing set I is totally ordered, and let $(e_\alpha)_{\alpha \in \mathbf{N}^{(I)}}$ be the basis introduced in the preceding no. We write $X_\alpha = f_E(e_\alpha)$ for $\alpha \in \mathbf{N}^{(I)}$. By (15) and (16), we have:

since $f_E$ is a cogebra morphism. We show that $f_E$ is injective. This results from the following lemma: