I'm starting to self-study differential geometry with Guillemin & Haine. They introduce a Lie derivative and in exercises they want to prove formula for lie bracket, i.e.:
Let $U\in \mathbb{R}^n$ be open and $v_1, v_2$ be vector fields on $U$. Show that there exists a unique vector field $w$ on $U$ with the property:
$$L_w(f) = L_{v_1}(L_{v_2}(f)) - L_{v_2}(L_{v_1}(f))$$
for all $f \in C^{\infty}$. Everything is fine until the vector fields are smooth, i.e.:
$$v_1 = \sum g_i \frac{\partial}{\partial x_i},$$ where $g_i$ are at least $C^1$ on $U$. I tried to find something but most of the answers explicitly assume vector fields to be smooth. Wikipedia alo seems to define Lie bracket for smooth vector fields. What's more in later exercises they indeed differentiate (partial derivatives) the $g_i$ functions without assuming they can. Yet - the definition o Lie derivative doesn't assume smoothness of vector fields.
My question is - is this just a gap in the exercise assumptions or is there a way to find the Lie bracket for non-differetiable vector fields?