In the book "Supergravity" written by Freedman & van Proeyen, a symplectic Majorana spinor is defined in eq. (3.86)

$$ \chi^i = \varepsilon^{ij} (\chi^j)^C, $$

where the upper index $C$ denotes charge conjugation and $\varepsilon^{ij} \varepsilon_{kj} = \delta^i_k$.

Below the above equation in the book, there is an exercise asking one to show that, in five dimensions, given the symplectic Majorana spinors $\psi^i$ and $\chi^i$, the quantity $\overline{\psi}^i \chi^j \varepsilon_{ji}$ is purely imaginary ($\overline{\psi}^i= (\psi^i)^T C$, with $C$ the charge conjugation matrix). I tried to show this last statement by doing the following computation

\begin{align} (\overline{\psi}^i \chi^j \varepsilon_{ji})^C &= (\overline{\psi}^i)^C (\chi^j)^C \varepsilon_{ji} \\ &=\varepsilon_{ki} \varepsilon_{lj} \overline{\psi}^k \chi^l \varepsilon_{ji} \\ & = \varepsilon_{lk}\overline{\psi}^k \chi^l \\ & = \overline{\psi}^i \chi^j \varepsilon_{ji}\,. \end{align}

Consequently, I get that $\overline{\psi}^i \chi^j \varepsilon_{ji}$ is real and not imaginary as the exercise states.

Is there a mistake in my computation? Or is the book statement that $\overline{\psi}^i \chi^j \varepsilon_{ji}$ should be imaginary wrong?

In the book "Supergravity" written by Freedman & van Proeyen, a symplectic Majorana spinor is defined in eq. (3.86)

$$ \chi^i = \varepsilon^{ij} (\chi^j)^C, \tag{3.86}$$

where the upper index $C$ denotes charge conjugation and $\varepsilon^{ij} \varepsilon_{kj} = \delta^i_k$.

Below the above equation in the book, there is an exercise asking one to show that, in 4+1 dimensions, given the symplectic Majorana spinors $\psi^i$ and $\chi^i$, the quantity $\overline{\psi}^i \chi^j \varepsilon_{ji}$ is purely imaginary ($\overline{\psi}^i= (\psi^i)^T C$, with $C$ the charge conjugation matrix). I tried to show this last statement by doing the following computation

\begin{align} (\overline{\psi}^i \chi^j \varepsilon_{ji})^C &= (\overline{\psi}^i)^C (\chi^j)^C \varepsilon_{ji} \\ &=\varepsilon_{ki} \varepsilon_{lj} \overline{\psi}^k \chi^l \varepsilon_{ji} \\ & = \varepsilon_{lk}\overline{\psi}^k \chi^l \\ & = \overline{\psi}^i \chi^j \varepsilon_{ji}\,. \end{align}

Consequently, I get that $\overline{\psi}^i \chi^j \varepsilon_{ji}$ is real and not imaginary as the exercise states.

Is there a mistake in my computation? Or is the book statement that $\overline{\psi}^i \chi^j \varepsilon_{ji}$ should be imaginary wrong?