I want to look at the transformation law for the commutator of two basis vectors. I understand completely how this works for a holonomic frame: The basis vector fields act on functions as the partial derivative operators wrt to the appropriate coordinates, so the commutator vanishes. But when we transform to an anholonomic frame, the basis vector (fields) aren’t going to be the partial derivatives for some coordinate system, so the resulting components will be nonvanishing.
This transformation law seems to me to reek of inhomogeneity, and I can’t seem to reconcile it with what I’ve been taught, which is that the commutator of vector fields is itself a vector field.
UPDATE: 
Here’s my problem. The object I have here should be a tensor (right?), and it vanishes in the holonomic frame. But now, in the anholonomic frame, I get nonzero terms. This should not be how a tensor transforms. What am I doing wrong here?
Since vectors transforms as $V^a=x^a_AV^A$ between two coordinate systems with $x^a_A:=\frac{\partial x^a}{\partial x^A}$,$$\begin{align}U^a\partial_aV^b&=U^Ax^a_A\partial_a(x^b_BV^B)\\&=U^A\partial_A(x^b_BV^B)\\&=U^A\partial_AV^Bx^b_B+U^AV^B\underbrace{x^b_{AB}}_{\tfrac{\partial^2x^a}{\partial x^A\partial x^B}},\end{align}$$so $U^a\partial_aV^b-V^a\partial_aU^b=x^b_B(U^A\partial_AV^B-V^A\partial_AU^B)$, i.e. $£_UV^b=x^b_B£_UV^B$. This is the usual transformation law for a vector.