  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 $\mathrm{Aut}_{\text{alg}}(X)$.
