To help me understand, I will try to type down two examples where I dont understand how it possibly can be like that, first of all, $d_{ij}$ is supposed to kronecker delta with $ij$ as indexes.
- $(d_{il}d_{jm} - d_{im}d_{jl})a_jb_lc_m$
- $(d_{kl}d_{im} - d_{km}d_{il})A_lC_mB_k$
I simply cant understand how I'm supposed to think, I thought $d_{il}A_l = A_i$.
Sorry for extremely hard to read, hopefully someone can help me, I cant understand how the two results works together. Thanks in advance!
As it comes from Einstein notation we summate over paired subscripts. For kronecker symbol we have that $d_{ij}=1$ if $i=j$ and equal to $0$ otherwise.
Thus, in application to the first expression:
$$ (d_{il}d_{jm} - d_{im}d_{jl})a_jb_lc_m=(d_{il}d_{jm}a_jb_lc_m)-(d_{im}d_{jl}a_jb_lc_m) $$ For the first term we take $m=j$, for the second $l=j$ $$ (d_{il}a_jb_lc_j)-(d_{im}a_jb_jc_m)$$ Now $l=i$ and $m=i$ respectively, which leads to $$a_jb_ic_j-a_jb_jc_i$$
Calculations for the second expression pretty much the same