When to relabel dummy variables in tensor index notation

Question

When writing the expression of $(\textbf a\times \textbf b)\bullet(\textbf c\times \textbf d) $ in index notation, I am told that the correct index notation is as follows:$$\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m$$ Why do we relabel the indices $j,k$ but not the index $i$? Is there a general rule for this?

Answer 1

You could relabel the index $i$ too, they're all mute. What you can't do is use the index $j$ and $k$ again for the second vector, because you're already using them in the first two summations. For one second, let's write summations again.

$$\begin{align*} (a \times b)\cdot (c \times d) &= \sum_i (a \times b)_\color{blue}{i} (c \times d)_\color{blue}{i} \\ &= \sum_i \left(\sum_{j,k}\epsilon_{\color{blue}{i}jk}a_jb_k\right)\left(\sum_{\color{red}{j,k}}\epsilon_{\color{blue}{i}\color{red}{jk}}c_\color{red}{j}d_\color{red}{k}\right) \\ &= \sum_{i,j,k,l,m}\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m.\end{align*}$$We still can write the step in red, but you need to relabel in the next step because $$\sum_{i,j,k,l,m}\epsilon_{ijk}a_jb_k\epsilon_{ilm}c_ld_m \neq \sum_{i,j,k}\epsilon_{ijk}a_jb_k\epsilon_{ijk}c_jd_k.$$As for relabeling $i$: $$\sum_i (a \times b)_i (c \times d)_i = \sum_p (a \times b)_p (c \times d)_p = \sum_q (a \times b)_q (c \times d)_q = \cdots$$