Definition. The commutator of two operations $A$ and $B$ is $[A,B] := ABA^{-1}B^{-1}$, or "do $A$, do $B$, undo $A$, undo $B$".
Example 0. In ${\mathbb Z}$, $[m,n]=0$.
Indeed, $2026+5-2026-5=0$.
Example 1. In $S_3$, $[(12), (23)] = (12) (23) (12)^{-1} (23)^{-1}=(123)$ and in general in $S_{\geq 3}$, \[ [(ij),(jk)]=(ijk). \]
Example 2. In $S_{\geq 4}$, \[ [(ijk), (jkl)] = (ijk) (jkl) (ijk)^{-1} (jkl)^{-1}=(il)(jk). \]
Example 3. In $S_{\geq 5}$, \[ [(ijk), (klm)] = (ijk) (klm) (ijk)^{-1} (klm)^{-1} =(jkm). \]
Example 4. So, in fact, in $S_5$, $(123) = [(412),(253)] = [[(341),(152)],[(125),(543)]]$ \[ = [[[(234),(451)],[(315),(542)]],[[(312),(245)],[(154),(423)]]] \] \[ = [\ [[[(123),(354)],[(245),(531)]],[[(231),(145)],[(154),(432)]]], \] \[ [[[(431),(152)],[(124),(435)]],[[(215),(534)],[(142),(253)]]] \ ], \] (and of course, we can go on like that as much as we want).