Commutator formula for powers of the Lefschetz operator ($[L^r,\Lambda ]{|}_{{\mathcal{A}}^k}=r(k-n+r-1)L^{r-1}$)

For the Hermitian Lefschetz operator on a complex $n$-manifold, the Lefschetz commutator and the commutator derivation identity imply this formula. When expanding a product commutator, one must evaluate $[L,\Lambda ]$ on the degree after the powers of $L$ to its right have acted.