  In this paper we show that totally geodesic subspaces determine the
commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge
4$. Many of the results are more general and apply to locally symmetric spaces
associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type
$B_n$ and $D_n$. We use a combination of techniques from algebraic groups and
quadratic forms to prove several results about these spaces.
