Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 3 / 10C / ii / Solution

The tensor contraction $B_{kl}={\delta }_{ij}{T}_{ijkl}$ is an isotropic second-rank tensor, because the Kronecker delta and $T$ are invariant under rotations. The classification proved above yields $B_{kl}=\mu {\delta }_{kl}$ for some scalar $\mu$.

For the given three-term tensor, contraction over $i,j$ givesThus $\mu =3\alpha +\beta +\gamma$.