How to compute Bochner laplacian $\Delta=\nabla^*\nabla=\sum \nabla_{e_i}$? I'm struggling with proving that Bochner laplacian can be described by the following formula similar to the standard laplacian formula from calculus: $$\Delta = \sum_i \nabla_i^2,$$ where $\nabla_i = \nabla_{e_i}$ for some local orthonormal basis $e_i$ and $\nabla$ is a connection satisfying any reasonable conditions (Levi-Civita on $TM$, the induced connections on $\Omega^k(M)$).
We live on a compact $C^\infty$ Riemannian manifold without boundary and the laplace operator is defined as $\nabla^*\nabla$, where $\nabla$ is viewed as a map $\Gamma(E)\to \Gamma(T^*M \otimes E)$ between spaces with scalar product and thus the adjoint operator $\nabla^*\!\!:\Gamma(T^*M \otimes E) \to \Gamma(E)$ is well defined*.
I'm new to Levi-Civita, so please be understanding if the solution is trivial...
*Actually the spaces here are neither finite dimensional nor Hilbert, so it is not clear if the adjoint operator exists... The explanation will be highly appreciated, but it can be assumed that $\nabla^*$ exists.

This calculation is done in full detail (without the assumption that the basis is orthonormal) in example $10.1.32$ of Liviu Nicolaescu's wonderful book Lectures on the Geometry of Manifolds which is freely available as a pdf from his website.
