Short notation for specific multiplication

Question

Say I have the product $A_iB_jC_kD_l$ which I know to be equal to $1$ iff $i=k$, $j=l$, and $i\ne j$. It equals $-1$ iff $i=l$, $j=k$, and $i\ne j$. And it equals $0$ otherwise. How do I write this in a compact notation? The $\pm 1$ or $0$ part reminds me of the Levi-Civita symbol, but I can't seem to think of a combination of Levi-Civita and Kronecker deltas that'll give me the right thing here.