Lie derivative of vectors and Lie derivative of 1-forms: can't reconcile the two

Question

I have been studying modern formulations of differential geometry, mostly by reading Flanders, Frankel and Sternberg. I think I have developed a working understanding of the Lie derivative of forms, and vectors, but I am having a hard time with the following paradox. Consider a set of basis vectors ${\bf e}_i$ on the manifold. The latter is equipped with a dot product, so that I can also consider the dual basis ${\bf e}^j=g^{ji}\bf{e}_i$. From the definition of the Lie derivative, $$ [{\bf e}_i,{\bf e}^j]={\cal L}_{{\bf e}_i}{\bf e}^j=g^{jm}[{\bf e}_i,{\bf e}_m]+{\bf e}_i(g^{mj}){\bf e}_m. (*) $$
In particular, if the ${\bf e}_i$'s commute then ${\cal L}_{{\bf e}_i}{\bf e}^j=0$ iff the $g^{ij}$'s are constant. Apply this to a holonomic basis ${\bf e}_i=\partial_i$. Then ${\bf e}^j=dx^j$. But if I consider $dx^i$ as a 1-form, and apply Cartan's magic formula $$ {\cal L}_{{\bf e}_i}dx^j=d\delta_i^j=0. (@) $$ This is what puzzles me. What I am doing wrong? Why are the (*) and (@) results different. Related to that, is the fact that for any vector field $$ {\cal L}_{\bf X}{\bf X}=0, $$ while, for the 1-form obtained applying the musical isomorphism to ${\bf X}$ in general $$ {\cal L}_{\bf X}X^\flat\neq 0. $$
Thank you for your time.

Answer 1

The point is the metric tensor is nonconstant from the viewpoint of the Lie derivative, unless $X$ is Killing. So this interferes when you try to push Lie derivative inside/outside the musical isomorphism.