= Symmetric-test criterion for a third-rank Cartesian tensor
{title2=$d_{ijk}s_{ij}\text{ is a vector for every symmetric }s\ \Longrightarrow\ d^s\text{ is a tensor}$}
If contraction of an array with every <symmetric second-rank tensor> is a <vector> in every orthonormal frame, its part symmetric in the contracted slots obeys the rank-three <tensor> transformation law. The difference between its proposed transformation and actual components is symmetric and contracts to zero against every symmetric <matrix>, so it vanishes. The antisymmetric part is invisible, by the <blindness of symmetric contraction tests to antisymmetric arrays>, and need not be tensorial.
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