Constructing an expression involving Kronecker delta and Levi-Civita symbol

Question

Let $i,j,k,l\in\{0,1,2\}$. I am looking for a simple expression $f(i,j,k,l)$ involving only (Kronecker delta) $$\delta_{ab}=\cases{1&if $a=b$\\0&else},$$ and (Levi-Civita symbol) $$\epsilon_{ab} = \cases{1&if $a<b$\\-1&if $a>b$\\0&if $a=b$},$$ and maybe constants, such that $f$ is non-zero if, and only if $\{i,j\}\neq\{k,l\}$. It would be perfect if the values taken by $f$ were only $\pm1$ or $0$, but it's not so necessary. Does anyone have one such expression under hand? Also, general techniques for constructing them would be greatly appreciated.

Answer 1

We have $$f(i,j,k,l)=\bigl(1-[ik][jl]\bigr)\bigl(1-[il][jk]\bigr),$$ where we denote $[ij]=\delta_{ij}$ for convenience.

Proof: Examine $[ik][jl]+[il][jk]$. It is one if either $(i,j)=(k,l)$ or $(i,j)=(l,k)$. It is two if both conditions are satisfied, which means all four numbers are equal. Whence $[ik][jl]+[il][jk]-[ik][jl][il][jk]$ is the negation of what you want.