I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using them for examples when computing cohomology with local coefficients). I was originally trained as a physicist, so I am also not very familiar with field theory (which comes up in the construction). So apologies in advance if there are some "dumb" questions here.
The purpose of my question is to understand the following construction referred to in the paper Local rigidity of hyperbolic manifolds with geodesic boundary by Kerckhoff and Storm, where they state:
"For each dimension $n>2$ [Gromov and Thurston] construct an infinite number of closed hyperbolic n-manifolds $V$ with the following properties: $V$ has a codimension 1 embedded, totally geodesic submanifold $M$ which itself has a codimension 1 embedded, totally geodesic submanifold $P$ (so $P$ is codimension 2 in $V$)."
The Gromov & Thurston paper being referred to is Pinching Constants for Hyperbolic Manifolds. It appears that the brief Section 1 contains the relevant construction, which I have snipped from the linked pdf:
I am unfortunately confused about "why" this construction gives us a hyperbolic manifold containing a codimension 1 totally geodesic hypersurface itself containing a totally geodesic codimension 1 hypersurface. To be more specific, why use this $\Phi_n$ instead of the usual Lorentzian quadratic form, and why finite index subgroups of this particular $\Gamma_n$?