# Trying to prove (a x (b x c)) = b (a.c) - c (a.b)

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.

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