Actually, there is an argument that doesn't use neither Weil identities nor flat up to a second order coordinates, instead, it uses the symplectic Hodge star, invented by Brylinski. When you have a 2n-dimensional manifold $M$ with the non-degenerate 2-form $\omega$, you can define an operator $*_s:\Lambda^k(M) \mapsto \Lambda^{2n-k}(M)$ by the standard identity $\alpha \wedge *_s\beta = \omega(\alpha,\beta)\omega^n$. Then, for a closed $\omega$ a really simple computation (you can do it only for the case of two variables, and then use induction on dimension) in Darboux coordinates shows that, up to some sign ($(-1)^{k+1}$, i guess), $*_sd*_s=[\Lambda, d].$ It is written in the Brylinski's article "A differential complex for Poisson manifolds". And then you need to observe that on a Kaehler $M$ symplectic and Riemannian Hodge stars differ by an action of $I$. The Kaehler identity $d^*=[\Lambda, d^c].$ follows from that.

Actually, there is an argument that doesn't use neither Weil identities nor flat up to a second order coordinates, instead, it uses the symplectic Hodge star, invented by Brylinski. When you have a 2n-dimensional manifold $M$ with the non-degenerate 2-form $\omega$, you can define an operator $*_s:\Lambda^k(M) \mapsto \Lambda^{2n-k}(M)$ by the standard identity $\alpha \wedge *_s\beta = \omega(\alpha,\beta)\omega^n$. Then, for a closed $\omega$ a really simple computation (you can do it only for the case of two variables, and then use induction on dimension) in Darboux coordinates shows that, up to some sign ($(-1)^{k+1}$, i guess), $*_sd*_s=[\Lambda, d].$ It is written in the Brylinski's article "A differential complex for Poisson manifolds". And then you need to observe that for a Kaehler $M$ symplectic and Riemannian Hodge stars differ by an action of $I$. The Kaehler identity $d^*=[\Lambda, d^c]$ follows from that.

Actually, there is an argument that doesn't use neither Weil identities nor flat up to a second order coordinates, instead, it uses the symplectic Hodge star, invented by Brylinski. When you have a 2n-dimensional manifold $M$ with the non-degenerate 2-form $\omega$, you can define an operator $*_s:\Lambda^k(M) \mapsto \Lambda^{2n-k}(M)$ by the standard identity $\alpha \wedge *_s\beta = \omega(\alpha,\beta)\omega^n$. Then, for a closed $\omega$ a really simple computation (you can do it only for the case two variables, and then use induction on dimension) in Darboux coordinates shows that, up to some sign ($(-1)^{k+1}$, i guess), $*_sd*_s=[\Lambda, d].$ It is written in the Brylinski's article "A differential complex for Poisson manifolds". And then you need to observe that on a Kaehler $M$ symplectic and Riemannian Hodge stars differ by an action of $I$. The Kaehler identity $d^*=[\Lambda, d^c].$ follows from that.

Actually, there is an argument that doesn't use neither Weil identities nor flat up to a second order coordinates, instead, it uses the symplectic Hodge star, invented by Brylinski. When you have a 2n-dimensional manifold $M$ with the non-degenerate 2-form $\omega$, you can define an operator $*_s:\Lambda^k(M) \mapsto \Lambda^{2n-k}(M)$ by the standard identity $\alpha \wedge *_s\beta = \omega(\alpha,\beta)\omega^n$. Then, for a closed $\omega$ a really simple computation (you can do it only for the case of two variables, and then use induction on dimension) in Darboux coordinates shows that, up to some sign ($(-1)^{k+1}$, i guess), $*_sd*_s=[\Lambda, d].$ It is written in the Brylinski's article "A differential complex for Poisson manifolds". And then you need to observe that on a Kaehler $M$ symplectic and Riemannian Hodge stars differ by an action of $I$. The Kaehler identity $d^*=[\Lambda, d^c].$ follows from that.