= Solution
For the <Lefschetz operator of a Kähler manifold> and its adjoint $\Lambda$, the <Kähler identities> include
$$
[\Lambda,\partial]=i\bar\partial^*,\qquad [\Lambda,\bar\partial]=-i\partial^*.
$$
They make the mixed anticommutators in the expansion of $\Delta_d$ vanish and imply $\Delta_\partial=\Delta_{\bar\partial}$. Expanding $d=\partial+\bar\partial$ therefore proves the <Kähler Laplacian identity>
$$
\boxed{\Delta_d=2\Delta_{\bar\partial}=2\Delta_\partial.}
$$
Thus $\Delta_d\alpha=0$ exactly when $\Delta_{\bar\partial}\alpha=0$.
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