  For any $k$-group scheme of finite type $G$ acting rationally on a
$k$-variety $X$, if the action is generically free then the dimension of
$\mathrm{Lie}(G)$ is upper bounded by the dimension of the variety. We show
that this is the only obstruction when $k$ is a perfect field and $G$ is
infinitesimal commutative trigonalizable. We also give necessary conditions to
have faithful rational actions of infinitesimal commutative trigonalizable
group schemes on varieties.
