Automorphism groups of affine varieties consisting of algebraic elements

Given an affine algebraic variety X X , we prove that if the neutral component A u t ∘<!-- ∘ --> ( X ) Aut^∘ (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 (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G G contains a closed connected nested ind-subgroup H ⊂<!-- ⊂ --> G H⊂ G , and for any g ∈<!-- ∈ --> G g∈ G some positive power of g g belongs to H H , then G = H G=H .