# About the realization of $SO(N)$ given by Daniel Bump in his book Lie Group

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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