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Let $X_M^c$ be the space of vector fields on a manifold $M$ with compact support. Prove that $X_M^c=[X_M^c,X_M^c]$.

Proving that $[X_M^c,X_M^c]\subset X_M^c$ is relatively simple. However, I have not been able to prove $X_M^c\subset [X_M^c,X_M^c]$. Any clues?

  • Is the problem easier/well known when $M$ is compact and you consider all vector fields? – Arnaud Mortier May 05 '18 at 01:04
  • @ArnaudMortier- Yes, I was thinking that in that case $X=[1,X]$. But $1$ is not a vector field....so maybe there's a clever vector field that essentially serves as a unit for this lie bracket operation.... –  May 05 '18 at 01:12
  • Have you worked it out explicitly for $M=\Bbb R$? – Ted Shifrin May 06 '18 at 00:38

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