  This paper shows that immersed totally geodesic $m$-dimensional suborbifolds
of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite
subgroups of the commensurator whenever $m \geqslant \lfloor \frac{n}{2}
\rfloor$. We call such totally geodesic suborbifolds finite centraliser
subspaces (or fc-subspaces) and use them to formulate an arithmeticity
criterion for hyperbolic lattices.
  We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has
infinitely many fc-subspaces, exhibiting examples of non-arithmetic orbifolds
that contain non-fc subspaces of codimension one. We provide an algebraic
characterisation of totally geodesically immersed suborbifolds of arithmetic
hyperbolic orbifolds by analysing Vinberg's commensurability invariants. This
allows us to construct examples with the property that the adjoint trace field
of the geodesic suborbifold properly contains the adjoint trace field of the
orbifold. The case of special interest is that of exceptional trialitarian
$7$-dimensional orbifolds. We show that every such orbifold contains a totally
geodesic arithmetic hyperbolic $3$-orbifold of exceptional type.
  Finally, we study arithmetic properties of orbifolds that descend to their
totally geodesic suborbifolds, proving that all suborbifolds in a
(quasi-)arithmetic orbifold are (quasi-)arithmetic.
