tensors - Solving antisymmetric tensorial equation

Assume we have the following Tensor objects: \begin{equation} F_{i}{}^{j}\;and\;S_{ij}{}^{k}, \end{equation} where the components of $F$ are known, and we would like to solve for the components of $S$ if they satisfy the following equation \begin{equation} F^{l}{}_{i}S_{jl}{}^{k}-F^{l}{}_{j}S_{il}{}^{k}=0. \end{equation} $l$ is summed over, all the indices run from 1 to 4, and $S$ is symmetric in the lower indices.

Can you please help in writing a Mathematica code for this.

My attempt:

First, suppose we know all the components of $F$, and they are given by \begin{equation} F= \begin{matrix} a & b & c & d\\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{matrix} \end{equation}

Then I defined the components of $S$ by:

S[i_, j_, k_] := S[i, j, k]

The first term of the equation I defined it as:

SF[i_, j_, k_] := SF[i, j, k] = S[1, i, j].F[k, 1] +
                                V[2, i, j].F[k, 2] + 
                                V[3, i, j].F[k, 3] + 
                                V[4, i, j].F[k, 4];

As for the second term in the equation, I think it can be found using Transpose

FS[i, j, k] = Transpose[SF, {i, k}]

Then for example:

Solve[SF==FS,{S[i,j,k]},{i,4},{j,4},{k,4}]

is not working. I'm sure there is something wrong in my commands, but I can't figure out what it is. The functions $a$,$b$,$c$,... in the expression of $F$ are some complicated scalar functions of space coordinates.

Answer

Perhaps

f = RandomInteger[{-1, 1}, {4, 4}]; 

Solve[
 And @@ Join[
   Thread[Equal[Flatten[Table[
       Sum[f[[l, i]] s[j, l, k] - f[[l, j]] s[i, l, k], {l, 4}], 
   {i, 4}, {j, 4}, {k, 4}], 2], 0]], 
   Flatten@Table[s[i, j, k] == s[j, i, k], {i, 4}, {j, 4}, {k, 4}]]]

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