Separability of unipotent-free abelian subgroups in linear groups

Let $\mathbf{F}$ be a field of characteristic zero. It is proved that for any finitely generated linear group $\mathrm{\Gamma }<{\mathsf{G}\mathsf{L}}_n(\mathbf{F})$, every unipotent-free abelian subgroup of $\mathrm{\Gamma }$ is separable.