TITLE: algorithm to compute the integral orthogonal group
QUESTION [10 upvotes]: Suppose I have an indefinite quadratic form over the integers, and I want to compute its orthogonal group. Is there an algorithm, or at least a heuristic? If yes, is there any implementation anywhere?

REPLY [3 votes]: I am currently writing such an algorithm (in Magma) for Lorentzian lattices. It is almost finished and has already been used to answer this question on MO.
It furnishes a fundamental domain for the action of the group of automorphisms $G$ of the lattice on the hyperbolic space, and identifies the boudary components. Thus it may be used to obtain a finite presentation of $G$.
Perhaps this algorithm will be available directly on Magma someday ... Now if you have an example in head, just tell me, I can perform the computations for you (if your example in not too twisted ... computations become quickly heavy).
Edit :
For your example : the Gram matrix is
A=  [0 1 1 1]
    [1 0 1 1]
    [1 1 0 1]
    [1 1 1 0] 

A fundamental domain is the convex hull of the following 4 points :
a=(0,0,1,1), b=(0,1,1,1), d=(1,1,1,1), c=(0,0,0,1)

(suitably rescaling a,b,c to make them fall in $\mathcal H_3$. The point c is a cusp.)
The reflections supported by the faces of this tetrahedron are 
[-1  1  1  1] (a,b,c)
[ 0  1  0  0]
[ 0  0  1  0]
[ 0  0  0  1]

[0 1 0 0] (a,c,d)
[1 0 0 0]
[0 0 1 0]
[0 0 0 1]

[1 0 0 0] (b,c,d)
[0 0 1 0]
[0 1 0 0]
[0 0 0 1]

[1 0 0 0] (a,b,d)
[0 1 0 0]
[0 0 0 1]
[0 0 1 0]

The matrices I gave in the comments below are conjugated to these.