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$\left(\begin{array}{rrrr|rrrr}1 & 1 & 1 & 1 & 1 & 0 & 0 & 0\\1 & 1 & -1 & -1 & 0 & 1 & 0 & 0\\ 1 & -1 & 1 & -1 & 0 & 0 & 1 & 0\\ 1 & -1 & -1 & 1 & 0 & 0 & 0 & 1\end{array}\right) = \left(\begin{array}{rrrr|rrrr}2 & 2 & 2 & 2 & 2 & 0 & 0 & 0\\0 & 0 & -2 & -2 & -1 & 1 & 0 & 0\\ 0 & -2 & 0 & -2 & -1 & 0 & 1 & 0\\ 0 & -2 & -2 & 0 & -1 & 0 & 0 & 1\end{array}\right) = \\ \left(\begin{array}{rrrr|rrrr}2 & 2 & 0 & 0 & 1 & 1 & 0 & 0\\0 & 0 & -2 & -2 & -1 & 1 & 0 & 0\\ 0 & -2 & 0 & -2 & -1 & 0 & 1 & 0\\ 0 & -2 & -2 & 0 & -1 & 0 & 0 & 1\end{array}\right) = \left(\begin{array}{rrrr|rrrr}2 & 2 & 0 & 0 & 1 & 1 & 0 & 0\\0 & 4 & 0 & 0 & 1 & 1 & -1 & -1\\ 0 & -2 & 0 & -2 & -1 & 0 & 1 & 0\\ 0 & -2 & -2 & 0 & -1 & 0 & 0 & 1\end{array}\right) = \\ \\ \left(\begin{array}{rrrr|rrrr}2 & 2 & 0 & 0 & 1 & 1 & 0 & 0\\0 & 2 & 0 & 0 & 1/2 & 1/2 & -1/2 & -1/2\\ 0 & -2 & 0 & -2 & -1 & 0 & 1 & 0\\ 0 & -2 & -2 & 0 & -1 & 0 & 0 & 1\end{array}\right) =\\ \left(\begin{array}{rrrr|rrrr}2 & 0 & 0 & 0 & 1/2 & 1/2 & 1/2 & 1/2\\0 & 2 & 0 & 0 & 1/2 & 1/2 & -1/2 & -1/2\\ 0 & 0 & 0 & -2 & -1/2 & 1/2 & 1/2 & -1/2\\ 0 & 0 & -2 & 0 & -1/2 & 1/2 & -1/2 & 1/2\end{array}\right)= \\ \left(\begin{array}{rrrr|rrrr}2 & 0 & 0 & 0 & 1/2 & 1/2 & 1/2 & 1/2\\0 & 2 & 0 & 0 & 1/2 & 1/2 & -1/2 & -1/2\\ 0 & 0 & 2 & 0 & 1/2 & -1/2 & 1/2 & -1/2\\ 0 & 0 & 0 & 2 & 1/2 & -1/2 & -1/2 & 1/2\end{array}\right)= \\ \left(\begin{array}{rrrr|rrrr}1 & 0 & 0 & 0 & 1/4 & 1/4 & 1/4 & 1/4\\0 & 1 & 0 & 0 & 1/4 & 1/4 & -1/4 & -1/4\\ 0 & 0 & 1 & 0 & 1/4 & -1/4 & 1/4 & -1/4\\ 0 & 0 & 0 & 1 & 1/4 & -1/4 & -1/4 & 1/4\end{array}\right) \\$