Assume that $M$ is a smooth $n$ dimensional manifold with $n>1$.
Is there a lie algebra structure on $\chi^{\infty} (M)$, the space of all smooth vector fields on $M$, such that we have the following property:
For every vector field $X\in \chi^{\infty}(M)$ the operator $ad_X:\chi^{\infty}(M) \to \chi^{\infty}(M)$ with $ad_{X}(Y)=[X,Y]$ is an elliptic differential operator of positive (non zero) order acting on the space of smooth vector fields on $M\setminus S$ where $S$ is the set of singular points of $X$.
Here is the MO link of this question
https://mathoverflow.net/questions/271336/the-adjoint-operators-as-elliptic-operators
EDIT:
According to answers to MO link, can one imagine a Lie structure on $\chi^{\infty}(M)$ whose adjoint operator would be a differential operator of order $d>1$?