*Ternary Quadratic Forms and Norms* edited by Olga Taussky (1982). Pages 5-30 is William Plesken, *Automorphs of Ternary Quadratic Forms*. The word automorph is one of the traditional terms for what would now be called a member of the (integer) automorphism group of the form, or rotation group, or orthogonal group.

Since $xz - y^2$ has the mixed coefficient $1,$ we need to double it to get an integral matrix, and this is also the Hessian matrix,

$$
H =
\left(
\begin{array}{rrr}
0 & 0 & 1 \\
0 & -2 & 0 \\
1 & 0 & 0
\end{array}
\right)
$$


This becomes annoying for diagonal forms, where we still double the diagonal entries:
$$
G =
\left(
\begin{array}{rrr}
38 & 0 & 0 \\
0 & 10 & 0 \\
0 & 0 & -2
\end{array}
\right)
$$

In this direction,
take 
$$
N =
\left(
\begin{array}{rrr}
38 & 30 & 16 \\
19 & -25 & -7 \\
-38 & 5 & 9
\end{array}
\right)
$$
for one of infinitely many solutions to
$$ N^T H N = -95 G  $$


give me a few more minutes...


In the other direction, take
$$
M =
\left(
\begin{array}{rrr}
2 & 2 & -2 \\
3 & -8 & -1 \\
11 & -4 & 9
\end{array}
\right)
$$
then
$$ M^T G M = -380 H  $$