= Solution
Adding and subtracting the transpose uniquely separates a <Cartesian second-rank tensor> into its <symmetric tensor> and <antisymmetric tensor> parts:
$$
\boxed{S_{ij}=\frac12(B_{ij}+B_{ji}),\qquad A_{ij}=\frac12(B_{ij}-B_{ji}).}
$$
Then $S_{ji}=S_{ij}$, $A_{ji}=-A_{ij}$ and $S_{ij}+A_{ij}=B_{ij}$. Both parts transform as tensors because transposition commutes with the orthogonal similarity $B'=QBQ^T$. Uniqueness follows by taking the symmetric and antisymmetric parts of any such decomposition.
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