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After we write the Levi-Civita symbol as being a determinant like we did previously mentioned, the fully antisymmetric residence that it have becomes evident: swapping any two indices corresponds to interchanging their corresponding rows in the matrix because of which the determinant, which can be the Levi-Civita by itself, alterations indicator.

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From this actions, we can certainly deduce what $epsilon^ ij4k $ could well be without the need of under-going the computations Again: it is just the results of exchanging the last two indices of $epsilon^ ijk4 $,

E α β γ δ = g α ζ g β η g γ θ g δ ι E ζ η θ ι . displaystyle E^ alpha beta gamma delta =g^ alpha zeta g^ beta eta g^ gamma theta g^ delta iota E_ zeta eta theta iota ,.

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