= Bracket and differential-form criteria for integrability
For an <almost complex structure>, closure of $T^{1,0}$ under the <Lie bracket> is equivalent to the absence of a $(0,2)$ component in $d\alpha$ for every $(1,0)$-form $\alpha$. Conjugation exchanges the bracket condition with closure of $T^{0,1}$. For sections $U,V$ of $T^{0,1}$, the <exterior derivative> identity gives $d\alpha(U,V)=-\alpha([U,V])$. Its vanishing for all $(1,0)$-forms is exactly $[U,V]\in T^{0,1}$.
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