= Solution
Complex conjugation preserves the real <Kähler form> and interchanges $\partial,\bar\partial$ and their formal adjoints. Conjugating the permitted identity among the <Kähler identities> therefore yields $[\Lambda,\bar\partial]=-i\partial^*$. The same nilpotence calculation as in part (a) gives $\bar\partial\partial^*+\partial^*\bar\partial=0$. Now expand $d=\partial+\bar\partial$ and $d^*=\partial^*+\bar\partial^*$:
$$
\Delta_d=\Delta_\partial+\Delta_{\bar\partial}
+\partial\bar\partial^*+\bar\partial^*\partial
+\bar\partial\partial^*+\partial^*\bar\partial.
$$
Both mixed anticommutators vanish, proving \b[$\boxed{\Delta_d=\Delta_\partial+\Delta_{\bar\partial}}$].
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