In my book the author makes the simplification$$(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}) \epsilon_{mni}=\epsilon_{jnl}-\epsilon_{ini}\delta_{jl}$$ My question is, without writing out all the sums, how do they motivate this?
$\delta $ = Kronecker-delta
$\epsilon$ = Levi-civita
Observe that you can further simply the formula, since $\epsilon_{ini}=0$. This is a direct consequence of the complete skew-symmetry of the Levi-Civita tensor $\epsilon_{ijk}$: $$ \epsilon_{ijk}=-\epsilon_{jik}=-\epsilon_{ikj}=-\epsilon_{kji}. $$ As for the rest, you need only to use repeatedly the following computation rule with the Kronecker $\delta$: for any tensor $A_{\ldots i\ldots}$: $$ \delta_{ik}A_{\ldots i\ldots}=\delta_{ki}A_{\ldots i\ldots}=A_{\ldots k\ldots} $$