Isotropic contractions of a fourth-rank tensor (${ϵ}_{ijk}{T}_{ijkl}=0,{\delta }_{ij}{T}_{ijkl}=\mu {\delta }_{kl},{ϵ}_{ijm}{T}_{ijkl}=\nu {ϵ}_{klm}$)

Contract an isotropic tensor with the Kronecker delta or the Levi-Civita symbol, using invariance under proper rotations. The three displayed contractions have ranks one, two and three. The only isotropic vector is zero; an isotropic second-rank tensor is a scalar delta, and an isotropic third-rank tensor is a scalar epsilon. For $T_{ijkl}=\alpha {\delta }_{ij}{\delta }_{kl}+\beta {\delta }_{ik}{\delta }_{jl}+\gamma {\delta }_{il}{\delta }_{jk}$, direct contraction gives $\mu =3\alpha +\beta +\gamma$ and $\nu =\beta -\gamma$.