Just an example. We begin with an isotropic ternary quadratic form,
$$ f(x,y,z) = 24 x^2 + 24 y^2 + 24 z^2 -43 yz - 43 z x - 43 x y. $$
Its Hessian matrix of second partial derivatives is
$$
H =
\left(
\begin{array}{rrr}
48 & -43 & -43 \\
-43 & 48 & -43 \\
-43 & -43 & 48
\end{array}
\right)
$$
Isotropic means that there is at least one triple $(x,y,z)$ of integers, not all equal to zero, with $f(x,y,z) = 0.$ Nondegenerate means $\det H \neq 0.$
The theorem in question guarantees an integer matrix,
$$
R =
\left(
\begin{array}{rrr}
58 & 61 & 18 \\
18 & -25 & 15 \\
15 & 55 & 58
\end{array}
\right)
$$
Back to that in a minute. The indicated form, $y^2 - z x,$ has Hessian matrix
$$
G =
\left(
\begin{array}{rrr}
0 & 0 & -1 \\
0 & 2 & 0 \\
-1 & 0 & 0
\end{array}
\right)
$$
Alright, the relationship is $R^T H R = 157339 G.$ That is,
$$
\left(
\begin{array}{rrr}
58 & 18 & 15 \\
61 & -25 & 55 \\
18 & 15 & 58
\end{array}
\right)
\left(
\begin{array}{rrr}
48 & -43 & -43 \\
-43 & 48 & -43 \\
-43 & -43 & 48
\end{array}
\right)
\left(
\begin{array}{rrr}
58 & 61 & 18 \\
18 & -25 & 15 \\
15 & 55 & 58
\end{array}
\right) =
\left(
\begin{array}{rrr}
0 & 0 & -157339 \\
0 & 314678 & 0 \\
-157339 & 0 & 0
\end{array}
\right)
$$
The rows of $R$ tell us that, for any integers $(u,v),$ we always have
$$ f( 58 u^2 + 61 uv + 18 v^2, 18 u^2 -25 uv + 15 v^2, 15 u^2 + 55 uv + 58 v^2 ) = 0 $$
