In index notation, the determinant of a 3 by 3 matrix $A$ is $\det(A)=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}$, and the product of matrices $A$ and $B$ is $(AB)_{ij}=A_{ik}B_{kj}$. Thus, the determinant of the product of matrices $A$ and $B$ is
$$\det(AB)=\epsilon_{ijk}A_{1l}B_{li}A_{2m}B_{mj}A_{3n}B_{nk}$$
On the other hand, the product of the determinants of matrices $A$ and $B$ is
$$\det(A)\det(B)=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}\epsilon_{lmn}B_{1l}B_{2m}B_{3n}$$
Since $\det(AB)=\det(A)\det(B)$, these expression are supposed to be equal. But (how) can I show the equation
$$\epsilon_{ijk}A_{1l}B_{li}A_{2m}B_{mj}A_{3n}B_{nk}=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}\epsilon_{lmn}B_{1l}B_{2m}B_{3n}$$
in index notation?
I have been able to show a few similar (although less complex) elementary relations nicely with index notation, but with this one I am out of my wits, and the only path I can see is manually writing out every single term.
