Adjoint of a commutator ($[A,B{]}^{\ast }=[B^{\ast },A^{\ast }]$)

When compositions and adjoints are defined on the common test domain, reversing products gives $[A,B{]}^{\ast }=[B^{\ast },A^{\ast }]$. In particular a self-adjoint Dolbeault Laplacian and the mutually adjoint Lefschetz operators satisfy $[{\Delta }_{\bar{\partial }},L{]}^{\ast }=[\Lambda ,{\Delta }_{\bar{\partial }}]$.