Lefschetz commutator ($[\Lambda ,L]{|}_{{\mathcal{A}}^k}=(n-k)\mathrm{id}$)

In complex dimension $n$, the Lefschetz operator of a Kähler manifold and adjoint Lefschetz operator satisfy $[\Lambda ,L]=(n-k)\mathrm{id}$ on degree $k$. In middle degree, this yields $\Vert L\alpha {\Vert }^2=\Vert \Lambda \alpha {\Vert }^2$.

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.