Quadratic minor null Lagrangian ($K_{k\gamma }{ϵ}_{\alpha \beta \gamma }{ϵ}_{ijk}{u}_{i,\alpha }{u}_{j,\beta }$)

For constant $K$, this quadratic combination of two-by-two gradient minors is a divergence. Indeed,because the term containing $u_{j,\beta \alpha }$ vanishes by antisymmetry of the Levi-Civita symbol. Its volume integral vanishes for homogeneous displacement boundary data. It also vanishes pointwise on rank-one matrices, since ${\nu }_{\alpha }{\nu }_{\beta }$ is symmetric. Consequently one can strengthen an elastic energy density by such a term without changing fixed-boundary incremental energies or acoustic tensors.