In Hutchings and Taubes's lecture note on Seiberg-Witten equation, a Weitzenbock formula is given, and the authors state that the Weitzenbock formula, proven in Donaldson's book using Kahler identities \begin{equation} {\bar \partial ^*} = i\Lambda \partial , \;\;\;\;\text{etc} \end{equation} are actually valid on a symplectic manifold with an adapted almost hermitian structure. Namely, it seems that Kahler identities do not require integrability of the almost complex structure.
I know the proof of Kahler identities on a Kahler manifold $X$: one first does it on $\mathbb{C}^2$, then choose a nice holomorphic coordinate on $X$, such that the metric $g$ on $X$ is just the Euclidean metric with $O(\Delta x^2)$ errors.
So I wonder:
Is there a similar trick (choosing a nice $(1,0)$ and $(0,1)$ frame?) for a symplectic manifold to prove Kahler identities? Any reference on how to actually prove them?