Symmetric-test criterion for a third-rank Cartesian tensor ($d_{ijk}{s}_{ij}\text{ is a vector for every symmetric }s⟹d^s\text{ is a tensor}$)

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.