Arithmetic trialitarian hyperbolic lattices are not LERF

A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in ${\mathbf{P}\mathbf{S}\mathbf{O}}_{7,1}(\mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that all arithmetic lattices in ${\mathbf{P}\mathbf{O}}_{n,1}(\mathbb{R})$, $n>3$, are not LERF.