Question on Einstein Summation Convention

Question

Question:

Solve for $\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k$

Attempted Solution:

$$\epsilon_{ij \ell} \, \epsilon_{km \ell} \, \epsilon_{ijm} \, a_k$$

$$\Rightarrow (\delta_{ik} \, \delta_{jm} - \delta_{im} \, \delta_{jk}) \epsilon_{ijm} \, a_k$$

$$\Rightarrow \delta_{ik} \, \delta_{jm} \, \epsilon_{ijm} \, a_k - \delta_{im} \, \delta_{jk} \epsilon_{ijm} \, a_k$$

$$\Rightarrow a_i \, \delta_{jm} \, \epsilon_{ijm} - a_j \, \delta_{im} \, \epsilon_{ijm}$$

However, how should I go about solving $a_i \, \delta_{jm} \, \epsilon_{ijm} - a_j \, \delta_{im} \, \epsilon_{ijm}$?

While I know the answer is supposed to be $0$, I am not sure how I can get from the above equation to the desired answer. Any help would be greatly appreciated!

Answer 1

I think the final quantity must be zero. The reason why is because of the terms $\delta_{jm}\epsilon_{ijm}$. In order for $\delta_{jm} = 1$ we must have $j = m$ which forces $\epsilon_{ijj}= 0$. Similarly, if all indices are different $\delta_{jm} = 0$.

Answer 2

Since $\delta_{jm}$ is symmetric but $\epsilon_{ijm}$ is $jm$-antisymmetric, their contraction is $0$. The other term vanishes for the same reason.