 Lefschetz commutator

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 $\Vert L\alpha {\Vert }^2=\Vert \Lambda \alpha {\Vert }^2$.