Solution
= Solution
Use three components of the proposed <isotropic tensor> form. Since $T_{1122}=0$, one gets $\alpha=0$. Since $T_{1212}=\epsilon_{123}^2=1$, one gets $\beta=1$. Finally $T_{1221}=\epsilon_{123}\epsilon_{213}=-1$ gives $\gamma=-1$. Hence <contraction of two Levi-Civita symbols> gives
$$
\boxed{\epsilon_{ijm}\epsilon_{klm}=\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk},\qquad(\alpha,\beta,\gamma)=(0,1,-1).}
$$