Modern proof of Lie's advective flow for Abelian groups I have been playing around with differential operators and came across Lie's advective flow for Abelian groups on Wikipedia here
$$
{\displaystyle e^{t~{\frac {\mathrm d}{\mathrm d h(x)}}}g(x)=e^{\frac{t}{h^\prime(x)}\frac{\mathrm d}{\mathrm dx}}g(x)=g(h^{-1}(h(x)+t)).}
$$
The Wikipedia articles cites a book by Lie (presumably with a proof of this identity); however, I am unable to read Norwegian and would like to read a proof of this identity.
Are there any references including a proof of this identity in English using modern notation?
 A: That's just a shrewd use of the shift operator. The idea is as follows : the operator $e^{t\beta(x)\partial_x}$ can be rewritten as $e^{t\partial_y}$ by the change of variable $y = h(x)$, where $h$ solves $h'(x) = \frac{1}{\beta(x)}$. Now, this operator is a shift operator for the variable $y$, i.e. $e^{t\partial_y}f(y) = f(y+t)$, hence the desired relation :
$$
e^{t\beta(x)\partial_x}g(x) = e^{t\partial_y}g(h^{-1}(y)) = g(h^{-1}(y+t)) = g(h^{-1}(h(x)+t))
$$

Addendum.
The problem is usually the reverse : one has a transformation $\phi(x)$ and we would like to find an exponential representation; in other words, we want $\beta(x) = \frac{1}{h'(x)}$, which necessitates first to solve the so-called Abel's equation $\phi(x) = h^{-1}(h(x)+1)$ for $h$ (see here), which is a very hard functional equation in general.
In fact, all these manipulations are based on the fact that the transformation $\phi$ belongs to an (abelian) group of functions $\phi_t(x) = h^{-1}(h(x)+t)$, which turn out to be the $t^\mathrm{th}$ iterates of $\phi$ (with a potentially non-integer $t$), and which is not guaranteed.
