I am confused about a claim on page 30 of the lectures notes of Hutchings and Taubes on Seiberg-Witten theory (https://math.berkeley.edu/~hutching/pub/tn.pdf).
Background: Let $M$ is a Riemannian 4-manifold endowed with a spin-c structure $\eta.$ Then we can consider the Clifford algebra bundle $\mathcal{P}= P_{\eta} \times_{Ad} (Cl(\mathbb{R}^4) \otimes \mathbb{C})$ and the complex spinor bundle $\mathcal{S}= P_{\eta} \times_{\rho} (Cl(\mathbb{R}^4)\otimes \mathbb{C}),$ where $\rho$ is the complex spin representation.
The bundle $\mathcal{P}$ acts on $\mathcal{S}$ by the complex spinor representation. Moreover, it follows from the construction of Clifford algebras that$\mathcal{P}$ is isomorphic as a vector bundle (not as a bundle of algebras) to the bundle of exterior differential forms. Hence it makes sense to consider the so-called "Clifford multiplication" of a complex spinor by a differential form.
Here is the statement I am struck on: $\beta \in \Omega^2(M)$ is a self-dual 2-form, then Clifford multiplication by $d\beta$ is the same as Clifford multiplication by $d^* \beta.$ I assume that I can check this by a very long computation (choosing a basis for the self-dual forms and explicitly computing the Clifford action of $d\beta$ and $d^*\beta$), but I assume that there is a more conceptual way to see this.