Coefficients and unordered placements

Slot symmetry and symmetry of $\mathfrak{s}$ imply ${\mathcal{K}}_{pq}={\mathcal{K}}_{qp}$. Let $\mathcal{N}$ be the set of unordered placements $\mathsf{k}=\{p,q\}$ with $|p|+|q|\le 2$, and put

There are $25$ unordered labels. This is an inventory of spanning labels, not a claim of linear independence of the scalar distributions.