I have been trying to show that: $ \vec{B}^{2} - \vec{E}^{2} =\frac{1}{2} f^{\mu \nu }f_{\mu \nu}$
where $\vec{B}^{2}$ and $\vec{E}^{2}$ are the square of the magnitude of the magnetic field and electric field respectively, and $f^{\mu \nu }$ is the electromagnetic tensor.
so far i have tried this:
$\vec{B}^{2} - \vec{E}^{2} = \frac{1}{4}(\varepsilon^{ijk}f_{jk} )(\varepsilon_{imn}f ^{mn} )- f^{0l}f_{l0} = \frac{1}{4}(\delta_{m}^{j} \delta_{n}^{k} - \delta_{n}^{j} \delta_{m}^{k})f_{jk}f ^{mn} + f^{l0}f_{l0}$
where i used the following properties
$B^{i} = \frac{1}{2}\varepsilon^{ijk}f_{jk}$
$\varepsilon_{imn} \varepsilon^{ijk} = (\delta_{m}^{j} \delta_{n}^{k} - \delta_{n}^{j} \delta_{m}^{k}) $
$f^{\mu \nu }=-f^{\nu \mu }$
then:
$=\frac{1}{4}(f_{mn}f^{mn}-f_{nm}f^{mn})+ f^{l0}f_{l0}= \frac{1}{2}f_{mn}f^{mn}+ f^{l0}f_{l0}$
what mistake did i make?
