Symmetric coefficients of a quadratic polynomial ($K^{ij}{p}_i{p}_j=K^{(ij)}{p}_i{p}_j$)

A quadratic homogeneous polynomial determines a unique symmetric bilinear coefficient tensor. An antisymmetric addition to its coefficient array gives zero contribution, so tensorial conditions on its coefficients must refer to this symmetric representative. In particular, the correspondence between quadratic geodesic first integrals and rank-two Killing tensors uses symmetric coefficients.