Given a general second tensor rank tensor A_(i j) and a metric g_(i j), define θ | congruent | A_(i j) g^(i j) = A_i^i ω^i | congruent | ϵ^(i j k) A_(j k) σ_(i j) | congruent | 1/2(A_(i j) + A_(j i)) - 1/3 g_(i j) A_k^k, where δ_(i j) is the Kronecker delta and ϵ^(i j k) is the permutation symbol. Then σ_(i j) + 1/3 θ g_(i j) + 1/2 ϵ_(i j k) ω^k = [1/2(A_(i j) + A_(j i)) - 1/3 g_(i j) A_k^k] + 1/3 A_k^k g_(i j) + 1/2 ϵ_(i j k)[ϵ^(λμ k) A_λμ] = 1/2(A_(i j) + A_(j i)) + 1/2(δ_i^λ δ_j^μ - δ_i^μ δ_j^λ) A_λμ = 1/2(A_(i j) + A_(j i)) + 1/2(A_(i j) - A_(j i)) = A_(i j), where θ, ω^i, and σ_(i j) are tensors of tensor rank 0, 1, and 2.
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