Embedability of right-angled Artin groups into hierarchically hyperbolic groups

For a hierarchically hyperbolic group, we provide sufficient conditions under which suitable powers of a finite collection of elements generate a right-angled Artin subgroup. Under additional hypotheses, we further show that this subgroup can be promoted to be quasi-isometrically embedded. Our framework recovers and unifies earlier results, including those of Clay-Leininger-Mangahas \cite{CLM12} and Runnels \cite{Run21} for mapping class groups, and of Kim-Koberda \cite{KK13} for right-angled Artin groups.