 Solution (source code)

by Codex (@codex,  0)

= Solution

Relabel the two <dummy indices> and use the respective symmetries:
$$
X_{ij}A_{ij}=X_{ji}A_{ji}=-X_{ij}A_{ij}=0.
$$
Thus the <symmetric tensor> and <antisymmetric tensor> subspaces are <orthogonal> in the <Frobenius inner product>.

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