Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator
$$
L = \sum_{i=1}^k X_i^2 + X_0~.
$$
Here, I assume that Hörmander's bracket condition is satisfied, that is to say that the Lie algebra generated by $X_0, X_1, \dots, X_k$ has full rank at every point in ${\mathbb R}^n$. This implies that $L$ is hypoelliptic. Now the question is: what are the conditions (if any) for the formal adjoint $L^\dagger$ of $L$ to be also hypoelliptic?

Judging by the answer to [this question](https://mathoverflow.net/questions/85578/when-the-adjoint-of-a-hypoelliptic-operator-hypoelliptic) it appears that this is trivially true since Hörmander's condition "does not change by taking adjoints". What this means is unclear to me. We have indeed that $$
L^\dagger = \sum_{i=1}^k X_i^2 - X_0 + f~,
$$
where $f$ is a smooth scalar function, right? If there were no $f$, then this would indeed be trivial, since we find again the same Lie algebra. Since all the treatments of Hörmander's theorem that I have seen discuss operators without such an $f$ (except Hormander's [paper](http://link.springer.com/article/10.1007%2FBF02392081) of 1967), I guess there is an easy way to get rid of this $f$. Do you know how? Or do you have another argument to show that $L^\dagger$ is also hypoelliptic?
Thanks a lot.