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 $[U,V]\in T^{0,1}$.