I think you are asking about repeated completing the square: as matrices, one such
$$ Q^T D Q = H $$
$$\left(
\begin{array}{rrr}
1 & 0 & 0 \\
\frac{ 1 }{ 4 } & 1 & 0 \\
\frac{ 1 }{ 4 } & - \frac{ 5 }{ 7 } & 1 \\
\end{array}
\right)
\left(
\begin{array}{rrr}
4 & 0 & 0 \\
0 & \frac{ 7 }{ 4 } & 0 \\
0 & 0 & \frac{ 13 }{ 7 } \\
\end{array}
\right)
\left(
\begin{array}{rrr}
1 & \frac{ 1 }{ 4 } & \frac{ 1 }{ 4 } \\
0 & 1 & - \frac{ 5 }{ 7 } \\
0 & 0 & 1 \\
\end{array}
\right)
= \left(
\begin{array}{rrr}
4 & 1 & 1 \\
1 & 2 & - 1 \\
1 & - 1 & 3 \\
\end{array}
\right)
$$
which can be revised (put all denominators into the diagonal matrix) to
$$ \frac{1}{4} \left(4x+y+z \right)^2 + \frac{1}{28} \left(7y-5z \right)^2 + \frac{13}{7} \left(z \right)^2 $$
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
$$ H = \left(
\begin{array}{rrr}
4 & 1 & 1 \\
1 & 2 & - 1 \\
1 & - 1 & 3 \\
\end{array}
\right)
$$
$$ D_0 = H $$
$$ E_j^T D_{j-1} E_j = D_j $$
$$ P_{j-1} E_j = P_j $$
$$ E_j^{-1} Q_{j-1} = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$
$$ H = \left(
\begin{array}{rrr}
4 & 1 & 1 \\
1 & 2 & - 1 \\
1 & - 1 & 3 \\
\end{array}
\right)
$$
==============================================
$$ E_{1} = \left(
\begin{array}{rrr}
1 & - \frac{ 1 }{ 4 } & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{1} = \left(
\begin{array}{rrr}
1 & - \frac{ 1 }{ 4 } & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{1} = \left(
\begin{array}{rrr}
1 & \frac{ 1 }{ 4 } & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{1} = \left(
\begin{array}{rrr}
4 & 0 & 1 \\
0 & \frac{ 7 }{ 4 } & - \frac{ 5 }{ 4 } \\
1 & - \frac{ 5 }{ 4 } & 3 \\
\end{array}
\right)
$$
==============================================
$$ E_{2} = \left(
\begin{array}{rrr}
1 & 0 & - \frac{ 1 }{ 4 } \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{2} = \left(
\begin{array}{rrr}
1 & - \frac{ 1 }{ 4 } & - \frac{ 1 }{ 4 } \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{2} = \left(
\begin{array}{rrr}
1 & \frac{ 1 }{ 4 } & \frac{ 1 }{ 4 } \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{2} = \left(
\begin{array}{rrr}
4 & 0 & 0 \\
0 & \frac{ 7 }{ 4 } & - \frac{ 5 }{ 4 } \\
0 & - \frac{ 5 }{ 4 } & \frac{ 11 }{ 4 } \\
\end{array}
\right)
$$
==============================================
$$ E_{3} = \left(
\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & \frac{ 5 }{ 7 } \\
0 & 0 & 1 \\
\end{array}
\right)
$$
$$ P_{3} = \left(
\begin{array}{rrr}
1 & - \frac{ 1 }{ 4 } & - \frac{ 3 }{ 7 } \\
0 & 1 & \frac{ 5 }{ 7 } \\
0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; Q_{3} = \left(
\begin{array}{rrr}
1 & \frac{ 1 }{ 4 } & \frac{ 1 }{ 4 } \\
0 & 1 & - \frac{ 5 }{ 7 } \\
0 & 0 & 1 \\
\end{array}
\right)
, \; \; \; D_{3} = \left(
\begin{array}{rrr}
4 & 0 & 0 \\
0 & \frac{ 7 }{ 4 } & 0 \\
0 & 0 & \frac{ 13 }{ 7 } \\
\end{array}
\right)
$$
==============================================
$$ P^T H P = D $$
$$\left(
\begin{array}{rrr}
1 & 0 & 0 \\
- \frac{ 1 }{ 4 } & 1 & 0 \\
- \frac{ 3 }{ 7 } & \frac{ 5 }{ 7 } & 1 \\
\end{array}
\right)
\left(
\begin{array}{rrr}
4 & 1 & 1 \\
1 & 2 & - 1 \\
1 & - 1 & 3 \\
\end{array}
\right)
\left(
\begin{array}{rrr}
1 & - \frac{ 1 }{ 4 } & - \frac{ 3 }{ 7 } \\
0 & 1 & \frac{ 5 }{ 7 } \\
0 & 0 & 1 \\
\end{array}
\right)
= \left(
\begin{array}{rrr}
4 & 0 & 0 \\
0 & \frac{ 7 }{ 4 } & 0 \\
0 & 0 & \frac{ 13 }{ 7 } \\
\end{array}
\right)
$$
$$ Q^T D Q = H $$
$$\left(
\begin{array}{rrr}
1 & 0 & 0 \\
\frac{ 1 }{ 4 } & 1 & 0 \\
\frac{ 1 }{ 4 } & - \frac{ 5 }{ 7 } & 1 \\
\end{array}
\right)
\left(
\begin{array}{rrr}
4 & 0 & 0 \\
0 & \frac{ 7 }{ 4 } & 0 \\
0 & 0 & \frac{ 13 }{ 7 } \\
\end{array}
\right)
\left(
\begin{array}{rrr}
1 & \frac{ 1 }{ 4 } & \frac{ 1 }{ 4 } \\
0 & 1 & - \frac{ 5 }{ 7 } \\
0 & 0 & 1 \\
\end{array}
\right)
= \left(
\begin{array}{rrr}
4 & 1 & 1 \\
1 & 2 & - 1 \\
1 & - 1 & 3 \\
\end{array}
\right)
$$