What does this notation means in noncommutative case

What does this notation means in noncommutative case

I've always used the covariant derivative in coordinate systems, therefore
Christoffel symbols were available, and moreover the coordinate basis
elements always commute, i.e. if $\{\partial x_i\}_{i=1,\dots,n}$ is a
coordinate basis, I've always have at hand the fact that $$[\partial
x_i,\partial x_j]=0.$$ Now I have to work with a non commutative basis, a
nonholonomic one to be precise. Then I looked at the wikipedia page for
the intrinsic formulas for the covariant derivative (I am not an expert
indeed), and the product rule there says that if $U,V$ are two vector
fields and $f$ is a function, then
$$\nabla_U(Vf)=f\nabla_UV+V(\nabla_Uf).$$ I'm pretty confident that if the
basis were commutative then $$\nabla_U(Vf)=f\nabla_UV+(\nabla_Uf)V.$$ What
can I say in the non commutative case?
Sorry if my question is stupid but I really am totally inexperienced in
differential geometry.
Best wishes
-Guido-