= Solution
Both the <Levi-Civita symbol> and $T$ are invariant under proper <rotations>. Their <tensor contraction>
$$
A_l=\epsilon_{ijk}T_{ijkl}
$$
is consequently an isotropic <vector>. The introductory proof shows that it must vanish, so \b[$\epsilon_{ijk}T_{ijkl}=0$].
For the displayed combination of <Kronecker deltas>, the three terms are
$$
\alpha\epsilon_{ijk}\delta_{ij}\delta_{kl},\qquad
\beta\epsilon_{ijk}\delta_{ik}\delta_{jl},\qquad
\gamma\epsilon_{ijk}\delta_{il}\delta_{jk}.
$$
In each term the <tensor contraction> repeats two indices of the <Levi-Civita symbol>; each term is zero. This verifies the identity for arbitrary $\alpha,\beta,\gamma$.
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