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Use three components of the proposed isotropic tensor form. Since $T_{1122}=0$, one gets $\alpha =0$. Since $T_{1212}={ϵ}_{123}^2=1$, one gets $\beta =1$. Finally $T_{1221}={ϵ}_{123}{ϵ}_{213}=-1$ gives $\gamma =-1$. Hence contraction of two Levi-Civita symbols gives

Use Einstein summation convention, writing ${\partial }_j=\partial /\partial x_j$. The contraction of two Levi-Civita symbols givesThe last equality is the product rule. Recognizing the divergences and directional derivatives proves the curl of a cross product identity: