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I am currently reading this paper. How can I prove (eq. 7 of the paper) that if we assume

$$ \chi ^{abcd} = \chi ^{cdab}$$

this leads to

$$ F \wedge \# F' = F' \wedge \# F $$

and vice versa?

$F$ is the electromagnetic 2-form and the Maxwell equations are given by

$$ d F = 0 \quad , \quad d \# F =J $$

where

$$ (\# F)_{ab} = \frac{1}{2} {\kappa _{ab}}^{cd}F_{cd} = \frac{1}{4}\epsilon_{abcd}\chi^{cdef}F_{ef} $$

where $\epsilon_{abcd}$ is the Levi-Civita symbol and

$$ \chi^{cdef} = \frac{1}{2}\epsilon ^{abcd} {\kappa_{cd}}^{ef} $$

with

$$ {\kappa_{ab}}^{cd} = - {\kappa_{ba}}^{cd} = - {\kappa_{ab}}^{dc} \quad , \quad \chi^{abcd} = -\chi^{bacd} = -\chi^{abdc}. $$

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