I was pondering on the symmetries that
\begin{equation}
\epsilon_{ijk}\delta_{\ell m}
\end{equation}
might have upon interchanging indices of the Kronecher delta and Levi-Civita symbol, e.g. the interchange of $i$ and $m$ or $\ell$.
Following this answer I get (see the proof below)
\begin{equation}
\epsilon_{ijk}\delta_{\ell m} = \epsilon_{\boldsymbol{m}jk}\delta_{\ell \boldsymbol{i}} + \epsilon_{\boldsymbol{\ell} jk}\delta_{\boldsymbol{i} m}
\end{equation}
This seems nice, and it's something I am looking for. But if I contract both sides with $\epsilon_{ijk}$ to make sure it holds well (sanity check!), I get the obvious contradiction
\begin{equation}
6\delta_{\ell m} = 2 \delta_{im}\delta_{i\ell} + 2 \delta_{i\ell} \delta_{im} = 4 \delta_{\ell m}
\end{equation}
What could possibly have gone wrong? Also, and more importantly, what would be the symmetries anyway?
Edit:
Proof:
Set $\hat x_i$ as the sum (see this answer)
$$\hat x_i=\delta_{i\ell}\hat x_\ell+\delta_{im}\hat x_m$$
(no summation is implied over repeated indices) so
$$\begin{align}
\epsilon_{ijk}\delta_{\ell m}&=\left(\hat x_i\cdot(\hat x_j \times \hat x_k)\right)\left(\hat x_{\ell} \cdot\hat x_m\right)\\\\
&=\left([\delta_{i\ell}\hat x_\ell+\delta_{im}\hat x_m]\cdot(\hat x_j \times \hat x_k)\right)\left(\hat x_{\ell} \cdot\hat x_m\right)\\\\
&=\delta_{i\ell}(\hat x_j \times \hat x_k)\cdot \hat x_{\ell}\hat x_{\ell}\cdot(\hat x_m)\\\\
&+\delta_{im}(\hat x_j \times \hat x_k)\cdot \hat x_{m}\hat x_{m}\cdot(\hat x_{\ell})\\\\
&=\delta_{i\ell}(\hat x_j \times \hat x_k)\cdot \hat x_m +\delta_{im}(\hat x_j \times \hat x_k)\cdot \hat x_{\ell}\\\\
&=\delta_{i\ell} \epsilon_{mjk} + \delta_{im}\epsilon_{\ell jk}
\end{align}$$
Best Answer
Let's first relabel the indices for better tracking: \begin{equation} \epsilon_{abc}\delta_{ij} \end{equation} Following this answer, let's assume none of the indices $(a,b,c)$ is equal to either of the other two, and hence the number $i$ is equal to one of $a$, $b$, or $c$. So, using the property of the Kronecker delta we can write
$$\hat x_i=\delta_{ia}\hat x_a+\delta_{ib}\hat x_b + \delta_{ic}\hat x_c$$
Hence, $$\begin{align} \epsilon_{abc}\delta_{ij}&=\left(\hat x_a\cdot(\hat x_b \times \hat x_c)\right)\left(\hat x_{i} \cdot\hat x_j\right)\\\\ &=[\hat x_a\cdot(\hat x_b \times \hat x_c)][(\delta_{ia}\hat x_a+\delta_{ib}\hat x_b + \delta_{ic}\hat x_c) \cdot\hat x_j)]\\\\ &=\delta_{ia}(\hat x_b \times \hat x_c)\cdot \hat x_{a}\hat x_{a}\cdot(\hat x_j)\\\\ &+\delta_{ib}(\hat x_c \times \hat x_a)\cdot \hat x_{b}\hat x_{b}\cdot(\hat x_{j})\\\\ &+\delta_{ic}(\hat x_a \times \hat x_b)\cdot \hat x_{c}\hat x_{c}\cdot(\hat x_{j})\\\\ &=\delta_{ia}(\hat x_b \times \hat x_c)\cdot \hat x_j + \delta_{ib}(\hat x_c \times \hat x_a)\cdot \hat x_j + \delta_{ic}(\hat x_a \times \hat x_b)\cdot \hat x_j\\\\ &= \epsilon_{jbc} \delta_{ia} + \epsilon_{ajc} \delta_{ib}+ \epsilon_{abj}\delta_{ic} \end{align}$$