  Given an affine algebraic variety $X$, we prove that if the neutral component
$\mathrm{Aut}^\circ(X)$ of the automorphism group consists of algebraic
elements, then it is nested, i.e., is a direct limit of algebraic subgroups.
This improves our earlier result. To prove it, we obtain the following fact. If
a connected ind-group $G$ contains a closed connected nested ind-subgroup
$H\subset G$, and for any $g\in G$ some positive power of $g$ belongs to $H$,
then $G=H.$
