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The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.

Is there a Riemannian manifold of dimension at least $2$ which satisfies either of the following conditions

1)The idealizer of the space of vector fields with vanishing divergence, properly contains this space

2)This space is an ideal in the space of all smooth vector fields

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    $\begingroup$ I lack deeper knowledge on this, but I remember reading somewhere that the diffeomorphism group is simple. I don't know if this is sufficient to say that the Lie algebra of all smooth vector fields is simple, but I would not be extremely surprised. $\endgroup$
    – Klaus Niederkrüger
    Commented Apr 10, 2019 at 22:49
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    $\begingroup$ What I wrote before is wrong: Take all vector fields that vanish on a chosen open subset $U\subset M$. This is an ideal, since the Lie bracket of any vector field with one of those will also vanish on the $U$. My apologies. $\endgroup$
    – Klaus Niederkrüger
    Commented Apr 11, 2019 at 13:34
  • $\begingroup$ @KlausNiederkrüger Thank you very much for your attention to my question and your comment. So you actualy answered the first part of my following question mathoverflow.net/questions/327214/… $\endgroup$
    – Ali Taghavi
    Commented Apr 12, 2019 at 19:12
  • $\begingroup$ I assume you mean the orientation preserving diffeomorphisms is a simple group, yes? $\endgroup$
    – Ali Taghavi
    Commented Apr 12, 2019 at 19:13

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