  The goal of this paper is to demonstrate the use of techniques from
hyperbolic geometry to compute generating sets of certain subgroups of
$SL^+(2,\mathbb{C})$; specifically, $SO^+(Q,\mathbb{Z})$ for $Q$ some integral
quadratic form of signature $(3,1)$ that does not represent 0. The algorithm is
illustrated for the form $Q_7=x_1^2+x_2^2+x_3-7x^4$, and explicit generating
matrices are found.
