 Solution

Both the Levi-Civita symbol and $T$ are invariant under proper rotations. Their tensor contractionis consequently an isotropic vector. The introductory proof shows that it must vanish, so ${ϵ}_{ijk}{T}_{ijkl}=0$.

For the displayed combination of Kronecker deltas, the three terms areIn each term the tensor contraction repeats two indices of the Levi-Civita symbol; each term is zero. This verifies the identity for arbitrary $\alpha ,\beta ,\gamma$.