# Charge conjugation of symplectic Majorana spinors in 4+1 dimensions

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?