What are recommended references for Lie algebra, with a focus on the calculus with Lie derivatives of functions?
I refer to the case, where the Lie derivative is reduced to the directional derivative. The following definition is from the lecture on Nonlinear systems of H. K. Khalil
https://www.egr.msu.edu/~khalil/NonlinearSystems/Sample/Lect_22.pdf
This is an excerpt:
$$\dot x= f(x)+g(x)u, \ \ \ \ y=h(x),$$ where $f$, $g$, and $h$ are sufficiently smooth in a domain $D$. $f :D\to\mathbb R^n$ and $g:D\to\mathbb R^n$ are called vector fields on $D$. $$\dot y=\frac{\partial h}{\partial x}[f(x)+g(x)u]=:L_fh(x)+L_gh(x)u.$$ $$L_fh(x) =\frac{\partial h}{\partial x}f(x)$$ is the Lie derivative of $h$ with respect to $f$ or along $f$.
I recommend Boothby's book for a mathematically-oriented, yet fairly clear presentation. Check out Isidori's book too if you need to see applications in nonlinear control.