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 Incoming links: Isotropic contractions of a fourth-rank tensor

C_{mk l} = ϵ_{ijm} T_{ijk l}

is an isotropic third-rank tensor under proper rotations. It therefore equals

ν ϵ_{mk l}

. Cyclically permuting three indices is even, so

ϵ_{mk l} = ϵ_{k l m}

and $ϵ_{ijm} T_{ijk l} = ν ϵ_{k l m}$ .

For the given Kronecker delta expression, the

α

term vanishes by antisymmetry, while the

β

and

γ

terms give respectively

β ϵ_{k l m}

and

γ ϵ_{l km} = - γ ϵ_{k l m}

. Hence

ϵ_{ijm} T_{ijk l} = (β - γ) ϵ_{k l m}, ν = β - γ .