It is known that a torsion free group can be simple (see e.g. 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.

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.

It is known that there a torsion free group can be simple (see e.g. 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.

It is known that a torsion free group can be simple (see e.g. 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.