$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Fixed $n \geq 2$, given $K \in \GL(n,Z)$. One can view $K$ is a Gram matrix of Lattice. I also imposed that $K$ is symmetric i.e $K^{T}=K$. We can define the automorphism group of $K$, $\Aut(K)=\{ W \in \GL(n,Z)\mathrel|W^TKW=K \}$. One can see a similar question at matrix congruence and smith normal form.
I know one can use Magma or Sage to compute the generator of $\Aut(K)$ if $K$ is positive definite (now $\Aut(K)$ is finite). But now I want to ask is there a way to compute some element of $\Aut(K)$? What I understand is that if $K$ is indefinite, then $\Aut(K)$ may be infinite. So I understand why both software can not compute the whole group. But what I just want to know: Is there a way at least compute to some of the elements in $\Aut(K)$?
To be more precise, given $K$ is indefinite, I know that $\pm I \in \Aut(K)$. But I want to know some other elements (I do not need to know all $\Aut(K)$). Where can I get such an algorithm or software? I have searched a lot but I still can not find such one.
Another question is that if I restricted some element of $W$ or some of the element $W$ satisfy some relation, can I check whether such $W \in Aut(K)$? I want to check whether such solution exists?
