# Is a polynomial group law on $\mathbb{R}^n$ automatically nilpotent?

I was told that a polynomial group law on (all of) $\mathbb{R}^n$ gives automatically a nilpotent (Lie, of course) group.

Is it true? Where can I find a proof?

A counterexample for open subsets of $\mathbb{R}^n$ is furnished by the halfplane with the $ax+b$ law.

• Torsten, make it an answer (not just a comment). Apr 17, 2010 at 19:13
• Moved a comment to an answer as per instructions. Apr 17, 2010 at 20:36