 Bochner-Weitzenbock formula for one-forms

For a real one-form $\alpha$, the Hodge Laplacian differs from the rough Laplacian by the action of Ricci curvature: $(\mathrm{Ric}\cdot \alpha )(X)=\alpha ({\mathrm{Ric}}^{♯}X)$, where $g({\mathrm{Ric}}^{♯}X,Y)=\mathrm{Ric}(X,Y)$. Equivalently, with the nonnegative scalar Laplacian convention,