= Solution
These are local criteria: multiplying by a smooth bump function extends any local test field or form to a global one without changing it near a chosen point. Assume the <Lie bracket> of any two sections of $T^{1,0}M$ again lies in $T^{1,0}M$. Because the <almost complex structure> is real, conjugating this condition shows that the <Lie bracket> of any two sections $U,V$ of $T^{0,1}M$ lies in $T^{0,1}M$.
For a $(1,0)$-form $\alpha$, the <exterior derivative> formula is
$$
d\alpha(U,V)=U(\alpha(V))-V(\alpha(U))-\alpha([U,V]).
$$
The first two terms vanish because $\alpha$ annihilates $T^{0,1}M$, and the last vanishes by bracket closure. The $(0,2)$ component is exactly the restriction of $d\alpha$ to two arguments in $T^{0,1}M$. Thus it is zero. Since the only types of a two-form are $(2,0)$, $(1,1)$ and $(0,2)$, \b[the bracket condition implies the stated differential-form condition]. This argument uses neither holomorphic coordinates nor an integrability theorem as an unproved intermediate step.
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