Exercise 9, § 1

Suppose that K is a $\mathbf{Q}$-algebra. Let $g$ be a Lie algebra admitting a K-basis and let U be its enveloping bigebra with the canonical filtration $(U_n)_{n \geq 0}$. Let $x \in U^+$ and let n be an integer $\geq 1$. Show that $x$ belongs to $U_n^+$ if and only if $c^+(x)$ belongs to $\sum_{i+j=n \atop i,j \geq 1} \mathrm{Im}(U_i^+ \otimes U_j^+)$.