The finite-subset criterion for nested ind-groups

Let $\mathfrak G$ be an ind-group, and let $X$ be a variety with automorphism group $\operatorname{Aut}(X)$. The finite-subset criterion. If every finite subset of $\mathfrak G$ (respectively, of $\operatorname{Aut}(X)$) is contained in an algebraic group, then $\mathfrak G$ (respectively, $\operatorname{Aut}(X)$) is nested. Here, nested means a union of an increasing sequence of algebraic subgroups.