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Thank for your interest for my question and thank you very much if you can answer me.

In his book p.187, Daniel Bump says that a realization (a representation) of $SO(2n)$ is given by the unitary matrice $g$ which verify $gJ^t g=J$ i.e. $J\,\overline{g}J=g$.

I understand why $$\begin{eqnarray} \left ( \begin{array}{cccccc} e^{-i y}&&\newline &e^{-i b}&\newline &&e^{2 i y}\newline &&&e^{-2 i y}&&&&& \newline &&&&e^{ib}&&&& \newline &&&&&e^{iy}&&& \newline \end{array} \right ) \end{eqnarray}$$ with eigenvalues $1,e^{3 i y/2}, e^{-3 i y/2},1,e^{3 i y/2}, e^{-3 i y/2}$ is similar to a matrix belonging to $SO(2n)$ but I don't understand why $$\begin{eqnarray} \left ( \begin{array}{cccccc} &&&e^{-i y}&&\newline &&&&e^{-i b}&\newline &&&&&e^{2 i y}\newline e^{-2 i y}&&&&& \newline &e^{ib}&&&& \newline &&e^{iy}&&& \newline \end{array} \right ) \end{eqnarray}$$ is similar to a matrix from $SO(2n)$ because its eigenvalues are $1,e^{3 i y/2}, e^{-3 i y/2},-1,-e^{3 i y/2}, -e^{-3 i y/2}$.

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