I have a hard time writing what I need an answer to, but ill e = the alternating tensoraoperator and $A_j$ is mean to be the $j$:th index to $A$. $\times$ is simply crossproduct.
I cant understand $$ (A \times (B \times C)) = e_{ijk} A_j e_{klm} B_l C_m = e_{kij} e_{klm} A_j B_l C_m $$ I cant understand the first step to the second, what is it with my tensors that makes it valid for me to "move in" $A_j$ among $B_l$ and $C_m$.
I understand its probably really tough to read what I've written, I simply don't know who to ask besides this forum. Appreciate any help!
The easy answer is that $A_j, B_l,C_m$ and $e_{ijk},e_{klm}$ are simply real (or complex) numbers, which are multiplied together (and then summed over as perscribed by the indices). The manipulation here is as valid as the manipulation $\sum_i x_iy_i = \sum_i y_ix_i$. The noncommutativity of the cross product is still evident in the fact that you keep all the indices the way they were, i.e. $A$ is still indexed by $j$ and so on.
Of course, we also use that $e_{ijk} = e_{kij}$, but that's not as mysterious.
Extra (because I should really be doing homework, but this is more fun): Written-out in classic vector notation, we have $$ [A_1,A_2,A_3]\times ([B_1,B_2,B_3]\times[C_1,C_2,C_3])\\ = [A_1,A_2,A_3]\times[B_2C_3 - B_3C_2,B_3C_1 - B_1C_3, B_1C_2 - B_2C_1]\\ = [\overbrace{\underset{e_{ijk}}+A_2(\underset{e_{klm}}+B_1C_2) } \overbrace{\underset{e_{ijk}}+ A_2(\underset{e_{klm}}-B_2C_1)} \overbrace{\underset{e_{ijk}}-A_3(\underset{e_{klm}}+B_3C_1)} \overbrace{\underset{e_{ijk}}- A_3(\underset{e_{klm}}-B_1C_3)},\ldots] $$ (where I skip the second and third component, because they don't help us). Each overbrace is one term of the summation $e_{ijk} A_j e_{klm} B_l C_m$, with $i = 1$ since this is the first component of the resulting vector. The manipulation they do in your question basically amounts to moving the minus sign in $A_2(-B_2C_1)$ to the front to make it $-A_2B_2C_1$ (and similarily for all other terms). Written like this it's hopefully not a big mystery that it's something you are allowed to do.