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As it is, why would anti-symmetric (0,2) tensor be traceless? Is it because trace should allow any variable for its indices?

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Let $A_{ij}$ be (the entries of) a totally antisymmetric tensor. Then, its trace is $g^{j i} A_{i j} = g^{i j} A_{i j} = -g^{i j} A_{j i} = - g^{j i} A_{i j} = 0$ because $g^{i j}$ is (the entries of) a totally symmetric tensor.