Cross Product Index Notation Proof - Do not understand step

Question

I am working through the document here:

Index Notation Notes

One thing that has confused me is the proof of the Tripe Cross Product on page 9 of this document.

Specifically, do not get how we go from the following line of the proof:

$\delta_{jh}\delta_{ki}a_ib_jc_k\hat{e}_h - \delta_{ji}\delta_{kh}a_ib_jc_k\hat{e}_h$

= $a_ib_jc_i\hat{e}_j - a_ib_ic_k\hat{e}_k$

Can someone please explain how these two are equal?

Answer 1

By definition of the delta notation, $\delta_{ki}c_k=c_i$ and similarly $\delta_{jh}\hat e_h=\hat e_j$. Substitute these into the first term and do the analogous in the second term to get the explanation.

Answer 2

Since $\delta_{jh}\hat{e}_h=\hat{e}_j$ while $\delta_{ki}c_k=c_i$, the first term on the left-hand side is $a_ib_jc_i\hat{e}_h$. Similarly, the term subtracted from it is $a_i\left(\delta_{ji}b_j\right)c_k\left(\delta_{kh}\hat{e}_h\right)=a_ib_ic_k\hat{e}_k$.