The proof of the classical Weitzenböck formula
$ \Delta (|f|^2)=|{\rm Hess}f|^2+\langle\nabla f, \nabla (\Delta f) +{\rm Ric} (\nabla f, \nabla f) \rangle $
uses the local orthonormal frame field $X_i$ around any fixed point $p\in M$ satisfy $ \langle X_i, X_j \rangle =\delta_{ij}, \ \ \nabla_{X_i}X_j(p)=0 $ to simplify the calculation.
My question is: What if I start with arbitary orthonormal fram say $\{e_1, \cdots, e_n\}$. My calculation shows that for any fixed $\alpha=1,\cdots,n$, the following holds: $ \begin{align} {\rm Hess}(|\nabla f|^2)(e_{\alpha}, e_{\alpha})= &2|\nabla f|^2 {\rm sec}(\nabla f, e_{\alpha}) + 2\nabla f \langle \nabla_{e_{\alpha}}\nabla f, e_{\alpha}\rangle +2 \langle \nabla _{e_{\alpha}}\nabla f, \nabla_{e_{\alpha}}\nabla f\rangle \\ &- 4\langle \nabla_{e_{\alpha}}\nabla f, \nabla_{\nabla f}e_{\alpha}\rangle \end{align} $ Where the ${\rm sec}$ denotes the sectional curvature spaned by $\nabla f$ and $e_{\alpha}$, .
The only difference between the standard calculation using normal fram and mine is the term $- 4\langle \nabla_{e_{\alpha}}\nabla f, \nabla_{\nabla f}e_{\alpha}\rangle $ So it means after summing up over $1, \cdots , n$, we must get $0$. i.e. $ \sum_{\alpha} - 4\langle \nabla_{e_{\alpha}}\nabla f, \nabla_{\nabla f}e_{\alpha}\rangle=0 $ But this seems not obvious to me. Did I miss something?
The classical calculation can be found here: The Comparison Geometry of Ricci Curvature, by Shunhui Zhu, 221-262 http://library.msri.org/books/Book30/contents.html