I would like to create a particular 28$\times$ 28 matrix whose entries are given by

$$ X_{[ij][kl]} = \delta^{[ij][kl]}_{1234} + \delta^{[ij][kl]}_{5678}$$

where $\delta^{ijkl}_{abcd} = \begin{cases} +1 & \text{if $ijkl$ is an even permutation of $abcd$}\\ -1 & \text{if ijkl is an odd permutation of $abcd$} \\0 & \text{otherwise}\end{cases}$

There are several tricky features about this matrix, making it hard to be implemented in Mathematica.

1. The entries of $X$ are labeled by antisymmetric pairs $[ij]$ where $i,j$ (individually) run from 1 to 8, but when they are combined in antisymmetric pairs, there are 28 combinations (here organised into 7 blocks, each 4$\times$ 4): (12, 34, 56, 78); (13, 24, 57, 68); (14, 23, 58, 67); (15, 26, 37, 48); (16, 25, 38, 47); (17, 28, 35, 46); (18, 27, 36, 45)
2. So basically, X is of block-diagonal form, with 7 blocks with entries labeled as above. So to calculate the entries, one uses 4-dimensional Levi-civita tensors for each entry. The only way I know to create this 28$\times$28 matrix would be to enter each entry individually using $Signature[\{i,j,k,l\}]$ function. This would repeat for the rest of the 6 remaining diagonal blocks (all entries corresponding to elements of different blocks vanish), and it is rather tedious.

So I'd be very grateful if someone could come up with a smart way to do this ?

I would like to create a particular 28$\times$ 28 matrix whose entries are given by

$$ X_{[ij][kl]} = \delta^{[ij][kl]}_{1234} + \delta^{[ij][kl]}_{5678}$$

where $\delta^{ijkl}_{abcd} = \begin{cases} +1 & \text{if $ijkl$ is an even permutation of $abcd$}\\ -1 & \text{if $ijkl$ is an odd permutation of $abcd$} \\0 & \text{otherwise}\end{cases}$

There are several tricky features about this matrix, making it hard to be implemented in Mathematica.

1. The entries of $X$ are labeled by antisymmetric pairs $[ij]$ where $i,j$ (individually) run from 1 to 8, but when they are combined in antisymmetric pairs, there are 28 combinations (here organised into 7 blocks, each 4$\times$ 4): (12, 34, 56, 78); (13, 24, 57, 68); (14, 23, 58, 67); (15, 26, 37, 48); (16, 25, 38, 47); (17, 28, 35, 46); (18, 27, 36, 45)
2. So basically, X is of block-diagonal form, with 7 blocks with entries labeled as above. So to calculate the entries, one uses 4-dimensional Levi-civita tensors for each entry. The only way I know to create this 28$\times$28 matrix would be to enter each entry individually using the Signature[{i,j,k,l}] function. This would repeat for the rest of the 6 remaining diagonal blocks (all entries corresponding to elements of different blocks vanish), and it is rather tedious.

So I'd be very grateful if someone could come up with a smart way to do this ?