When is the automorphism group of an affine variety nested?

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