Automorphism groups of affine varieties consisting of algebraic elements

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 $G=H.$