When is the automorphism group of an affine variety nested?

Abstract: For an affine algebraic variety $X$, we study the subgroup $\mathrm{Aut}{\text{alg}}(X)$ of the group of regular automorphisms $\mathrm{Aut}(X)$ of $X$ generated by all the connected algebraic subgroups. We prove that $\mathrm{Aut}{\text{alg}}(X)$ is nested, i.e., is a direct limit of algebraic subgroups of $\mathrm{Aut}(X)$, if and only if all the $\mathbb{G}a$-actions on $X$ commute. Moreover, we describe the structure of such a group $\mathrm{Aut}{\text{alg}}(X)$.