Hi! I'm trying to understand a proof for the Bochner-Weitzenbock formular. I'm sorry I have to bother you with such a basic question but I've worked at this for more than an hour now, but I just don't get the very first step, i.e.:

Where we are in a complete Riemannian manifold, [itex]f \in C^\infty(M)[/itex] at a point [itex]p \in M[/itex], with a local orthonormal frame [itex]X_1, ..., X_n[/itex] such that [itex]\langle X_i, X_j \rangle = \delta_{ij}, D_{X_i}X_j(p) = 0[/itex], and of course

[itex]\langle \nabla f, X \rangle = X(f) = df(X)[/itex]
[itex]\textrm{Hess }f(X, Y) = \langle D_X(\nabla f), Y \rangle[/itex]
[itex]\Delta f = - \textrm{tr(Hess )}f[/itex]

I've tried to use the Levi-Civita identities, but I'm getting entangled in these formulas and don't get anywhere.