= Solution
Define
$$
\boxed{s_{ij}=\frac{t_{ij}+t_{ji}}2,\qquad a_{ij}=\frac{t_{ij}-t_{ji}}2.}
$$
They sum to $t_{ij}$, with $s_{ji}=s_{ij}$ and $a_{ji}=-a_{ij}$. Interchanging two slots of a <tensor> still produces a <tensor>, so these are a <symmetric second-rank tensor> and an <antisymmetric second-rank tensor>, not merely symmetric and antisymmetric arrays in a selected frame.
For uniqueness, if $t=s+a=s'+a'$, then $s-s'=a'-a$. The common difference is both symmetric and antisymmetric, so each of its components equals its own negative and vanishes over $\mathbb R$. Thus \b[the decomposition is unique]. These are the <symmetric and antisymmetric parts of a matrix>, with their <tensor> transformation law preserved.
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