**What is an interesting class of examples of hyperbolic 3-manifolds,
each of which satisfies the following conditions?**

**1. It is compact**

**2. Its trace field contains a unique imaginary quadratic extension.**

**3. Its quaternion algebra is isomorphic to one of the form $\Big(\frac{a,b}{K}\Big)$, where $a,b\in K\cap\mathbb{R}$.**

Since the word 'interesting' is not well-defined, I'll settle for any examples, but it would be cool if they have some nice combinatorial or geometric characterization.

Then there is a follow-up question (more for algebraic number theorists).
**How could I replace conditions 2 and 3 with something in terms of the algebra's ramification set?** That is, there is a number field $F=K\cap\mathbb{R}$ so that the algebra is isomorphic to $\Big(\frac{a, b}{F(\sqrt{-d})}\Big)$
where $a,b\in F$, and $d\in F^+$.
But yet it is still a division algebra (else the manifold is most likely not compact).
Can this be (at least for some $F, d$ choices) phrased in terms of the divisors of $a$ and $b$?

I expect one would need to already know the definitions involved to answer this, but I will supply the arithmetic ones below for the sake of readers. Afterall the more people who know this, the more people I have to talk to!

To a hyperbolic 3-manifold $M$
is associated a Kleinian group $\Gamma\cong\Pi_1(M)$
represented in $\mathrm{PSL}_2(\mathbb{C})$.
The *trace field* of $M$
is the field $\mathbb{Q}(\{\mathrm{tr}(\gamma)\mid\gamma\in\Gamma\})$,
which I'll denote by $k_0 M$.
Using the character variety and algebraic geometry, it follows that this is a number field (a finite extension of $\mathbb{Q}$),
and by Mostow rigidity it is a manifold invariant.

Using the same setup,
the *quaternion algebra*
of $M$
is defined as
$$\big\{\sum_{i=t}^nt_i\gamma_i\mid t_i\in k_0M,\gamma_i\in\Gamma,n\in\mathbb{N}\big\}$$
and is commonly denoted by $A_0M$.
A quaternion algebra is a 4-dimensional central-simple algebra,
and it can be proven that $A_0M$
is such a thing using the Skolem-Noether theorem.
This is a stronger manifold invariant.
These algebras (provided the field is not characteristic 2, which doesn't matter here since we're using number fields) necessarily take the following form. If $K$ is the field it is over, then the algebra looks like
$$K\oplus Ki\oplus Kj\oplus Kij$$
where $i^2=a, j^2=b, ij=-ji$, with $a,b\in K\setminus\{0\}$.
And we denote this by the Hilbert symbol $\Big(\frac{a,b}{K}\Big)$.
A property of these algebras is that they are identified up to isomorphism by ramification of their places (field embeddings and prime ideals) over $K$.
Quaternion algebras of non-compact manifolds never have ramification over their primes. Quaternion algebras of compact manifolds typically do have ramification of their primes, but there are some strange examples where they don't.