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How to prove \begin{align} F_{\mu\rho} \tilde{F}_{\nu}^{\phantom{\nu}\rho} = \frac{1}{4} \eta_{\mu\nu} F_{\rho\sigma} \tilde{F}^{\rho \sigma} \end{align} using Schouten identity \begin{align} 0 = 5 \delta_{\mu}^{[\nu } \epsilon^{\rho\sigma\tau\lambda]} = \delta_{\mu}^{\phantom{\mu}\nu} \epsilon^{\rho\sigma\tau\lambda} +\delta_{\mu}^{\phantom{\mu}\rho}\epsilon^{\sigma\tau\lambda\nu} +\delta_{\mu}^{\phantom{\mu}\sigma}\epsilon^{\tau\lambda\nu\rho} +\delta_{\mu}^{\phantom{\mu}\tau}\epsilon^{\lambda\nu\rho\sigma} +\delta_{\mu}^{\phantom{\mu}\lambda} \epsilon^{\nu\rho\sigma\tau} \end{align} Where the dual tensor is defined in this way \begin{align} \tilde{F}^{\mu\nu} = -\frac{1}{2} i \epsilon^{\mu\nu\alpha\beta} F_{\alpha\beta} \end{align}


My guess \begin{align} &R.H.S=\frac{1}{4} \eta_{\mu\nu} F_{\rho\sigma} \tilde{F}^{\rho \sigma} = -\frac{1}{8} i \eta_{\mu\nu} \epsilon^{\rho\sigma\alpha\beta} F_{\rho\sigma} F_{\alpha\beta} \\ &L.H.S=F_{\mu\rho} \tilde{F}_{\nu}^{\phantom{\nu}\rho} = F_{\mu\rho} \tilde{F}^{x\rho}\eta_{x \nu} = -\frac{1}{2} i \epsilon^{x \rho \alpha \beta} \eta_{x \nu} F_{\mu \rho} F_{\alpha\beta} \\ & \eta_{\mu\nu} \epsilon^{\rho\sigma\alpha\beta} F_{\rho\sigma} F_{\alpha\beta} = 4\epsilon^{x \rho \alpha \beta} \eta_{x \nu} F_{\mu \rho} F_{\alpha\beta} \quad (?) \end{align} To prove the last line i think Schouten identity is needed.


Due to below answer (@Noiralef) Explicit computations are following

Rewrite the schouten identity by pulling down $\nu$ \begin{align} 0 = \eta_{\mu\nu} \epsilon^{\rho\sigma\tau\lambda} +\delta_{\mu}^{\phantom{\mu}\rho}\eta_{x\nu}\epsilon^{\sigma\tau\lambda x} +\delta_{\mu}^{\phantom{\mu}\sigma}\eta_{x\nu}\epsilon^{\tau\lambda x\rho} +\delta_{\mu}^{\phantom{\mu}\tau}\eta_{x\nu}\epsilon^{\lambda x\rho\sigma} +\delta_{\mu}^{\phantom{\mu}\lambda}\eta_{x\nu} \epsilon^{x \rho\sigma\tau} \end{align} Then \begin{align} &\left( \eta_{\mu\nu} \epsilon^{\rho\sigma\tau\lambda} +\delta_{\mu}^{\phantom{\mu}\rho}\eta_{x\nu}\epsilon^{\sigma\tau\lambda x} +\delta_{\mu}^{\phantom{\mu}\sigma}\eta_{x\nu}\epsilon^{\tau\lambda x\rho} +\delta_{\mu}^{\phantom{\mu}\tau}\eta_{x\nu}\epsilon^{\lambda x\rho\sigma} +\delta_{\mu}^{\phantom{\mu}\lambda}\eta_{x\nu} \epsilon^{x \rho\sigma\tau} \right) F_{\rho\sigma}F_{\tau\lambda} \\ &=\eta_{\mu\nu} \epsilon^{\rho\sigma\tau\lambda}F_{\rho\sigma}F_{\tau\lambda} + \epsilon^{\sigma\tau\lambda x} \eta_{x\nu} F_{\mu\sigma}F_{\tau\lambda} + \epsilon^{\tau\lambda x\rho} \eta_{x\nu} F_{\rho\mu}F_{\tau\lambda} + \epsilon^{\lambda x \rho\sigma}\eta_{x\nu} F_{\rho\sigma}F_{\mu\lambda}+ \epsilon^{x \rho\sigma\tau} \eta_{x\nu}F_{\rho\sigma} F_{\tau\mu} \\ & = \eta_{\mu\nu} \epsilon^{\rho\sigma\tau\lambda}F_{\rho\sigma}F_{\tau\lambda} + 4\epsilon^{\sigma\tau\lambda x} \eta_{x\nu} F_{\mu\sigma}F_{\tau\lambda} =0 \\ & \quad \Rightarrow \quad \eta_{\mu\nu} \epsilon^{\rho\sigma\tau\lambda}F_{\rho\sigma}F_{\tau\lambda} =- 4\epsilon^{\sigma\tau\lambda x} \eta_{x\nu} F_{\mu\sigma}F_{\tau\lambda} = 4\epsilon^{x \sigma\tau\lambda } \eta_{x\nu} F_{\mu\sigma}F_{\tau\lambda} \end{align} Here we used \begin{align} \epsilon^{\tau\lambda x \rho} F_{\rho\mu} F_{\tau\lambda} = \epsilon^{\tau\lambda x \sigma} F_{\sigma\mu} F_{\tau\lambda} = (-1)^3 \epsilon^{\sigma\tau\lambda x} F_{\sigma\mu} F_{\tau\lambda} = \epsilon^{\sigma\tau\lambda x} F_{\mu\sigma} F_{\tau\lambda} \end{align}

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Rewrite the Shouten identity by pulling down $\nu$: $$ 0 = \eta_{\mu\nu} \epsilon^{\rho\sigma\tau\lambda} + \delta_\mu^\rho \eta_{\chi\nu} \epsilon^{\sigma\tau\lambda\chi} + \cdots$$ Then plug this into the left hand side of the equation you marked with (?). You will get four terms, all having the correct structure of the right hand side (one free index at $\eta$ and one at one of the $F$'s, all other lower indices are contracted with the $\epsilon$).

Now you only have to check that everything works out and those terms have the same signs, so you arrive at the right hand side.

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