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What is a sufficient condition for a lie subgroup $G$ of $GL(n,\mathbb{R})$ to be the automorphism group of a Lie structure on $\mathbb{R}^{n}$. In particular does $O(n)$ satisfies this property?

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    $\begingroup$ What do you mean by 'Lie structure'? Do you just mean the structure of a Lie algebra on $\mathbb{R}^n$? $\endgroup$ Mar 15, 2014 at 14:15
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    $\begingroup$ If you did mean the structure of a Lie algebra on $\mathbb{R}^n$, then the answer is 'yes' for $n=3$ and $G = \mathrm{SO}(3)$ (but 'no' for $\mathrm{O}(3)$ when $n=3$) and no for $G =\mathrm{SO}(n)$ and $\mathrm{O}(n)$ all other $n$. $\endgroup$ Mar 15, 2014 at 14:28
  • $\begingroup$ @RobertBryant Could you please introduce a reference(book) for the reasons of the statements which you mentioned. $\endgroup$ Mar 15, 2014 at 20:00

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