= Solution
\b[False.] Tests against <symmetric second-rank tensors> see only the symmetric part $A_s=(A+A^T)/2$, because the <Frobenius inner product> of a symmetric matrix and an antisymmetric matrix is zero:
$$
\sum_{ij}K_{ij}B_{ij}=-\sum_{ij}K_{ji}B_{ij}=-\sum_{ij}K_{ij}B_{ij}=0.
$$
An arbitrary basis-dependent antisymmetric part is invisible to these tests. For example, prescribe
$$
A=\begin{pmatrix}0&1&0\\-1&0&0\\0&0&0\end{pmatrix}
$$
in one basis, and $A'=0$ in another rotated basis, choosing antisymmetric arrays in every remaining basis. Every contraction with any <symmetric second-rank tensor> is the invariant scalar zero. But an invertible rotation cannot transform this nonzero matrix to zero, so the array is not a <Cartesian second-rank tensor>. This is the <blindness of symmetric contraction tests to antisymmetric arrays>.
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