I am struggling to reconcile two views of the Lie bracket $\mathbf{[X, Y]}$ of two vector fields $\mathbf{X}$ and $\mathbf{Y}$.
I know that the Lie bracket measures the failure of the flow in the successive directions $\mathbf{X}$, $\mathbf{Y}$, $-\mathbf{X}$, $-\mathbf{Y}$ to return to the point $\mathbf{x}$, see Wikipedia: Lie bracket of vector fields.
This view is consistent with, for example, the vector fields of the "unit" polar coordinates $$X = \frac{x}{r} \partial_x + \frac{y}{r} \partial_y$$ $$Y = -\frac{y}{r} \partial_x + \frac{x}{r} \partial_y$$ where $r := \sqrt{x^2+y^2}$. The Lie bracket works out to be $\mathbf{[X, Y]} = -\frac{1}{r} Y$.
At the point $(x,y)^T=(1,0)^T$, for example, the Lie bracket evaluates to $(0,-1)^T$. This matches the following picture that follows the flows:
But, using the shift operator along the integral curves of the vector fields, flowing first with $\mathbf{X}$ and then with $\mathbf{Y}$ transports $\mathbf{P}$ to $\mathbf{A}$: $$x^i(A) = \exp(\varepsilon Y) \exp(\varepsilon X) x^i |_P$$
Starting again at $\mathbf{P}$ and reversing the sequence of flows yields point $\mathbf{B}$: $$x^i(B) = \exp(\varepsilon X) \exp(\varepsilon Y) x^i |_P$$
Hence, using the Taylor series of the exponential function, the vector pointing from $\mathbf{A}$ to $\mathbf{B}$ is $$x^i(B) - x^i(A) = \varepsilon^2 \mathbf{[X, Y]}$$ to the lowest order in $\varepsilon$.
This seems reasonable to me, yet the visualization is:
And the red gap vector points, in this interpretation, in the positive $y$ direction. But we have already established that it points in the negative $y$ direction (both pictorially and calculating the Lie bracket).
The view of a loop of flows is correct (and matches the example), yet using the shift operator seems correct to me, too, but there is a mismatch (in sign?) to the corresponding image.
Can someone give me a hint on where the reasoning with the exponential operators goes wrong?