Suppose that $q$ is an integral quadratic form, not necessarily unimodular or even non-degenerate, and write $O_q(\mathbb{Z})$ for its automorphism group. According to this question this group is finitely generated, but I gather that writing down a list of generators isn't so easy.
Suppose that $q$ is strongly indefinite, in the sense that its rank is large compared to its signature. Are generators known in this case? More precisely, let $q_n$ be the orthogonal direct sum of $q$ with $n$ copies of the hyperbolic form $H$ of rank $2$. Are generators (not necessarily a finite list) of $O_{q_n}(\mathbb{Z})$ known?
In the case of unimodular forms, generators are known by the work of Wall (On the orthogonal groups of unimodular quadratic forms. II. Journal für die reine und angewandte Mathematik 213 (1963/4), 122 - 136.) The simple form of these generators makes them very useful for applications to $4$-manifolds, which is my motivation for this question. If the assumption that $q$ is non-degenerate simplifies matters, I'd be interested in the answer to the question in that situation.
