Let $X$
be a complete finite-volume orientable hyperbolic $3$-manifold,
and let $\Gamma$
be a Kleinian representation of $\pi_1(X)$.
Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big)$,
i.e. its *trace field*.
Let $\Gamma^{(2)}:=\langle\gamma^2\mid\gamma\in\Gamma\rangle$,
the group generated by squares in $\Gamma$
and let $k\Gamma=K\Gamma^{(2)}$,
i.e. its *invariant trace field*.
Then these are number fields,
$K\Gamma$
is a manifold invariant and $k\Gamma$
is a commensurability invariant.

I've been told that if $X$ contains a immersed closed totally-geodesic surface, then $K\Gamma$ is a degree $2$ extension of a real field $F$ (where $F$ is the trace field of the surface under a fixed embedding into $X$).

Is this correct? Do they really mean to say this of $k\Gamma$ rather than of $K\Gamma$? Are there some additional conditions I'm missing that do make this true for $K\Gamma$?

If this is true, would you please provide a reference? (I'm scouring through my old copy of Maclachlan and Reid and can't seem to find it.) If it isn't true, would you know a counterexample?

**Edit:**
I think this is true if I add the requirement that $\Gamma$
be arithmetic. The only part of that I'm missing is that the trace field of the surface must be the full real part of the trace field of the $3$-manifold. I can't find a reference for that either though.

It's obvious (trivial even) in the arithmetic noncompact case.