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!