The anti-symmetry property of the magnetic field tensor, $\mathbf{F}{ij} = -\mathbf{F}{ji}$, is used to show the symmetry of the product $\mathbf{M}{ik} = \mathbf{F}{ij} \mathbf{F}{jk}$ by demonstrating that $\mathbf{M}{ik}$ is equal to its transpose, $\mathbf{M}_{ki}$.
Here is the step-by-step application:
Define the Transpose: Start with the transpose of the tensor product, $\mathbf{M}_{ki}$:
$$ \mathbf{M}{ki} = \mathbf{F}{kj} \mathbf{F}_{ji} $$
Apply Anti-Symmetry: Substitute the anti-symmetry relation ($\mathbf{F}{ab} = -\mathbf{F}{ba}$) to both terms in the transpose:
Substituting these gives:
$$ \mathbf{M}{ki} = (-\mathbf{F}{jk})(-\mathbf{F}_{ij}) $$
Simplify and Reorder: The two negative signs multiply to a positive sign, and because $j$ is a dummy summation index, the order of multiplication of the scalar terms can be changed:
$$ \mathbf{M}{ki} = \mathbf{F}{jk} \mathbf{F}{ij} = \mathbf{F}{ij} \mathbf{F}_{jk} $$
Conclusion: Since the transpose $\mathbf{M}{ki}$ is shown to be equal to the original tensor $\mathbf{M}{ik}$ (where $\mathbf{M}{ik} = \mathbf{F}{ij} \mathbf{F}{jk}$), the tensor product $\mathbf{F}{ij} \mathbf{F}_{jk}$ is symmetric in the free indices $i$ and $k$.