How does alkene metathesis involving Schrock carbenes proceed through a four-membered ring intermediate?

In alkene metathesis, Schrock carbenes—which feature a metal-carbon double bond polarized with the carbon as the electron-rich center—facilitate the reaction by interacting with an alkene.1

The process proceeds through a metallacyclobutane intermediate, which is a four-membered ring containing the metal atom and the carbon atoms from the carbene and the alkene. In this mechanism, the metal-carbene complex undergoes a [2+2] cycloaddition with the alkene to form this four-membered ring. Subsequently, the ring undergoes a [2+2] cycloreversion, breaking the bonds in a different orientation to release a new alkene and generate a new metal-carbene species. This cycle allows for the exchange of alkylidene groups between molecules, effectively rearranging the double bonds of the organic reactants.12