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I'm trying to prove that $$ \mathbf{ a }\times (\mathbf{b} \times \mathbf{c}) = \mathbf{b} ( \mathbf{a} \cdot \mathbf{c} ) - \mathbf{c} ( \mathbf{a} \cdot \mathbf{b} ), $$ using the identity $$ \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km}. $$

This is how I've progressed so far: $$ a \times (b \times c) = \epsilon_{ijk} a_j [b \times c]_k $$ $$ = \epsilon_{ijk} \epsilon_{klm} a_j b_l c_m $$ $$ = (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl})a_j b_l c_m $$

I know the next step is meant to be $$ = b_i a_j c_j - c_i a_j b_j $$

with the following step proving the original claim $$ =b_j ( \mathbf{a} \cdot \mathbf{c} ) - c_i ( \mathbf{a} \cdot \mathbf{b}). $$

I understand how the epsilon identity works and I think I have applied it correctly. However, I don't understand how to get from $ (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl})a_j b_l c_m $ to $ b_i a_j c_j - c_i a_j b_j $

I've been scouring lecture material and can't find an explanation of how to use the delta identity and so I just can't comprehend the jump.

Any help would be appreciated, thank you.

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Multiplying an expression $E_{\cdots j \cdots}$ by $\delta_{ij}$ (and summing over repeated indices, which we always assume here) allows you to replace the index $j$ with $i$. (Why? Because when you let the summed over index $j$ run over all possible values, only the case of $j=i$ contributes a non-zero amount.)

So for example $$ \delta_{il} (\delta_{jm} a_jb_lc_m) = \delta_{jm} a_jb_ic_m= b_i \delta_{jm} a_j c_m = b_i a_m c_m = b_i \mathbf{a\cdot c} $$

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