Let $X,Y$ be complete smooth vector fields on, say, $\mathbb{R}^2$.
What would be an example that $[X,Y]$ is not complete?
2026-03-27 06:13:20.1774592000
Example that $[X,Y]$ is not complete
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$[x^2\partial_y,y\partial_x]=x^2\partial_x-2xy\partial_y$. The outcome is incomplete say because it's tangent to the line $y=0$, where it becomes $x^2\partial_x$, whose flow is easy to compute and see that it's not complete. (I guess there are easier examples, but it's the best I could do :)