The adjoint operator as elliptic operator

Question

Is there a  Lie  algebra  structure on the  space  of  smooth vector fields on a  manifold for  which the "Adjoint" operator would  be  an elliptic operator?

I  explained (and asked) this  question in the  following MO  link but I did not  get a perfect and  complete  answer.

The  motivation for this  question is  explained in the  following  MO  link. The  motivation comes  from a  talk in Toulouse  and  and arxived preprint.

https://mathoverflow.net/questions/271336/the-adjoint-operators-as-elliptic-operators

Thank you