Let $(X, g_x)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms over $X$. Then the Hodge-star decomposes $\Lambda^2$ into the space of self-dual and anti-self-dual 2-forms $$\Lambda^2 = \Lambda^2_+ \oplus \Lambda^2_- .$$ Assume that there exists a non-vanishing section of $\Lambda^2_+$, say $\omega$. Then $\omega$ defines an almost-complex structure $I$ over $X$. Consider the complexification with respect to the almost-complex structure: $$ \Lambda^2 \otimes \mathbb{C} = \Lambda^{2,0} \oplus \Lambda^{0,2} \oplus \Lambda^{1,1}. $$

Then $$ \Lambda^2_+ = \Lambda^{2,0} \oplus \Lambda^{0,2} \oplus \mathbb{C}\cdot \omega~~\text{and}~~ \Lambda^2_- = \Lambda^{1,1}_{0} $$ where, $\Lambda^{1,1}_{0}$ is the space orthogonal to the space spanned by $\omega$ in $\Lambda^{1,1}$.

Let $\mathcal{Z}$ denote the twistor space of $X$ which is just the sphere bundle inside $\Lambda^2_+$. We can write $\omega$ as a real-valued function times a section of $\mathcal{Z}$ $$ \omega = |\omega |\cdot \left(\frac{\omega}{|\omega|}\right):= |\omega|\cdot \omega', ~~ \omega' \in \Gamma(\mathcal{Z}). $$

My question is the following:

Consider $\nabla \omega$, where $\nabla$ is the Levi-Civita connection. In four dimensions $\nabla \omega \in \Lambda^{1,2} \oplus \Lambda^{2,1}$, the first component being the Nijenhuis tensor and the second one $d\omega$. However, $$ d \omega = d|\omega| \wedge \omega' + |\omega|\cdot d\omega'. $$

The first term contains both $(2,1)$ and $(1,2)$-components, contradicting the fact that $d\omega \in \Lambda^{2,1}$. Am missing something here?