composition product) of differential operators ${\mathbb{D}}(M)$. It is true that the first-order elements of both are the vector fields, thus coincide. However, we are going to show that if first order differential operators are to be considered primitive elements, then ${\mathbb{D}}(M)$ cannot be given a natural Hopf algebra structure; we thank P. Aschieri for making us aware of the following argument, since published [@Corfu]. Consider the linear map ${\mathcal{L}}:{\mathcal{U}}({\mathfrak{X}}(M))\to{\mathbb{D}}(M)$ obtained by extending the Lie derivative. This map is not onto because zeroth order differential operators are functions, whereas the zeroth order elements of ${\mathcal{U}}({\mathfrak{X}}(M))$ are just scalars. Actually, ${\mathbb{D}}(M)$ is a ${\mathcal{F}}(M)$-module, while ${\mathcal{U}}({\mathfrak{X}}(M))$ is not. For this very reason ${\mathbb{D}}(M)$ cannot be a Hopf algebra. Consider two commuting linearly independent vector fields $X,Y$ nonvanishing on a common domain (e.g. locally let $X$ be the partial derivative ${\partial}_i$ and $Y$ a different one ${\partial}_j$), and the vector fields $aX,aY$, where $a$ is an arbitrary function, nonvanishing on the same domain. The composition $a{\mathcal{L}}_X{\mathcal{L}}_Y = a{\mathcal{L}}_Y{\mathcal{L}}_X$ is an element in ${\mathbb{D}}(M)$. Suppose *arguendo* that there exist on ${\mathbb{D}}(M)$ a coproduct ${\delta}$ compatible with composition of operators, and such that vector fields are primitives. We would have $$\begin{aligned}
{\delta}(a{\mathcal{L}}_X{\mathcal{L}}_Y) &= a{\mathcal{L}}_X{\mathcal{L}}_Y {\otimes}1 