I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to implement some software using a naive approach, but the results have been less than fruitful. As an example 


Let $K = {\mathbb Q}(I)$ (where $I^2=-1$) and let 
$$Q = \{ z\_1 + z\_2 i + z\_3j + z\_4k\;|\;z\_i\in K,\;i^2 = 2,\;j^2=5,\;ij= k\}$$
Also let 
$$ D = \{ z\_1 + z\_2 i + z\_3j + z\_4k\in Q\;|\; q\text{ has reduced norm 1},\; z\_i\in \mathbb Z[I]\} $$
According to general theory, $Q$ is a skew field and the elements $D$ form a cocompact lattice in $\text{SL}(2,{\mathbb C})$. So by a paper by A. Borel and Harish-Chandra, one can find a finite number of generators for this group $D$ (and in general, any cocompact lattice).


Any advice on finding generators for cocompact lattices in general would be hugely helpful. Thanks in advance!



Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices
This is precisely what my [Magma package](http://www.normalesup.org/~page/software.html) does, if the base field $K$ has at most one complex place. The algorithm is described in [this paper](http://arxiv.org/abs/1206.0087).


In the example you give, the set $\{ -7 + (-4I + 5)i + (24I + 6)j + (17I + 4)k, -2I + 3Ij + 2Ik, 3 + Ii + 4Ij + 3Ik, 2 - 4Ii + 11Ij + 8Ik, -I - Ii, -I + 2i - 4j - 3k, 2 + Ii - 7Ij - 5Ik, -3 - Ii - 10j - 7k, -6 + 13j + 9k, -3I - 4j - 3k, (-4I + 1) + (2I + 4)i + (6I - 14)j + (4I - 10)k, 4I + Ii + (10I + 7)j + (7I + 5)k, I + 2i + 10Ij + 7Ik, -4I + 3i + 13Ij + 9Ik, -I - 3i + 14j + 10k, (4I - 1) + (-2I - 4)i + (6I - 14)j + (4I - 10)k, (-4I - 1) + (-2I + 4)i + (-6I - 14)j + (-4I - 10)k \}$ generates the group $D$.


