Equivalence of two definitions of Laplace-Beltrami on differential forms

I know of two ways of defining the (negative - depending on your convention) Laplace-Beltrami operator on the differential forms of a compact, orientable Riemannian manifold $$M$$.

1. The Levi-Civita connection extends to a connection tensor bundles by Leibniz rule $$\nabla(a\otimes b)=\nabla a\otimes b + (-1)^aa\otimes\nabla b$$ (and similarly for wedges) and by defining it on $$1$$-forms by $$(\nabla\alpha)(X,Y) = \nabla_X\alpha(Y)-\nabla_Y\alpha(X)-\alpha([X,Y])$$ (is this correct?). In particular, we have a connection $$\nabla:\Omega^k(M)\to\Gamma(M,T^*M\otimes\Lambda^kM)$$ and another $$\nabla:\Gamma(M,T^*M\otimes\Lambda^kM)\to\Gamma(M,T^*M^{\otimes 2}\otimes\Lambda^kM)$$. We can now concatenate them and take the negative of the trace with respect to the metric $$\Delta=-tr_g(\nabla\nabla).$$
2. Using Hodge theory, we can define $$\Delta=-(dd^\star+d^\star d)$$.

Is there an easy way to see whether these two definitions are equal (possibly without computing in coordinates)? A reference where it is done would be awesome!

• They’re not the same. See en.wikipedia.org/wiki/Weitzenb%C3%B6ck_identity Apr 1, 2023 at 21:58
• @Deane Aha, that explains why I couldn't prove it... If you write your comment as an answer (if possible with a small description of the identity and a reference?) I will be very happy to accept it! Apr 2, 2023 at 10:23