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Let $L$ be a Lie algebra and denote by $\mathrm{Der}(L)$ the derivation algebra of $L$. If $L$ is finite-dimensional, then a theorem of Leger and Togo (Duke Math. J. $\bf{26}$ (1959), 623 – 628, DOI: 10.1215/S0012-7094-59-02660-2) states that $\mathrm{Der}(L)$ is nilpotent if and only if $L$ is characteristically nilpotent or 1-dimensional. I recall that a Lie algebra is said to be characteristically nilpotent if all of its derivations are nilpotent linear transformations.

Is there any result of this sort in the infinite-dimensional case? What can be said on the structure of $L$ when $\mathrm{Der}(L)$ is nilpotent?

For an infinite dimensional Lie algebra $L$ admitting a nonnilpotent (outer) derivation, is it possible that $\mathrm{Der}(L)$ is a nilpotent Lie algebra?

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  • $\begingroup$ 1) You haven't defined "characteristically nilpotent", I see 2 possible meanings, in terms of automorphisms, and in terms of derivations (which are equivalent for finite-dimensional Lie algebras in characteristic zero) 2) you haven't specified the ground field, possibly you have in mind the complex field, or arbitrary characteristic... please could you fix this? $\endgroup$
    – YCor
    Commented Apr 8, 2016 at 13:07
  • $\begingroup$ Yves: The definition of "characteristically nilpotent" is already included, and I am interested in Lie algebras defined over arbitrary fields. $\endgroup$
    – Salvatore Siciliano
    Commented Apr 8, 2016 at 14:35
  • $\begingroup$ Well, probably a direct sum of finite-dimensional characteristically nilpotent Lie algebras of nilpotency length tending to infinity can yield a characteristically nilpotent Lie algebra (in the "pointwise" meaning you give) that is not nilpotent modulo its center (and hence has non-nilpotent inner derivation algebra). (This would disprove one implication of the equivalence you mention, but not in the direction of your last question) $\endgroup$
    – YCor
    Commented Apr 8, 2016 at 14:39
  • $\begingroup$ OK, thanks. What happens by using the definition of characteristically nilpotent Lie algebra as in the survey of Ancochea-Campoamor instead of the pointwise one? Of course, by the Engel-Jacobson Theorem, in the finite-dimensional case these definitions are equivalent, but in the infinite dimensional case the pointwise one is weaker. $\endgroup$
    – Salvatore Siciliano
    Commented Apr 8, 2016 at 15:00
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    $\begingroup$ I just realized that if the derivations of L are nilpotent of bounded index, then Der(L) clairly satisfies an Engel condition and so it is locally nilpotent (indeed nilpotent in case of characteristic 0) by a celebrated result of Zelmanov. $\endgroup$
    – Salvatore Siciliano
    Commented Apr 10, 2016 at 15:52

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