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It is known that a torsion-free group can be simple (see e.g. Rataggi's paper https://www.degruyter.com/abstract/j/jgth.2007.10.issue-3/jgt.2007.028/jgt.2007.028.xml). I would like to know if the situation in the context of modular Lie algebras is similar. More precisily:

Let $\mathbb{F}$ be a field of characteristic $p>0$. Does there exist a restricted simple Lie algebra over $\mathbb{F}$ with no nonzero $p$-algebraic element?

I recall that an element $x$ of a restricted Lie algebra $(L,[p])$ is said to be $p$-algebraic if the restricted subalgebra generated by $x$ is finite-dimensional.

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  • $\begingroup$ Your link is for torsion-free finitely presented simple groups. For arbitrary groups, it's much easier. For instance it's been known for decades that the group of self-homeomorphisms of $\mathbf{R}$ with compact support is simple. Also torsion-free finitely generated simple groups were constructed in the 50s. $\endgroup$ – YCor Jun 17 '19 at 19:27
  • $\begingroup$ Thank you for the remark. Are you aware of any similar results for restricted Lie algebras? $\endgroup$ – Nathan Jun 17 '19 at 20:45
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    $\begingroup$ You already asked this question in the post! $\endgroup$ – YCor Jun 18 '19 at 7:50

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