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Let $L\neq 0$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. If $F$ is algebraically closed, then it is known that $L$ has no nontrivial restricted subalgebras if and only if $L$ is 1-dimensional.

Over arbitrary fields of positive characteristic, is there any description of restricted Lie algebras with the previous property?

It is clear that $L$ is generated (as a restricted Lie algebra) by a single element $z$. But what can be said about $z$?

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This is only a partial answer. For an element $x$ of $L$, denote by $\langle x \rangle_p$ the restricted subalgebra generated by $x$. Since $L=\langle z \rangle_p$, $L$ is finite-dimensional and so the element $z$ is $p$-algebraic. Moreover, either $z^{[p]}=0$ or $z$ is a semisimple element. In fact, if $\dim L>1$ then we must have $\langle z^{[p]}\rangle_p=\langle z \rangle_p$. However, if $z$ is semisimple, in general $L$ need not be free of nonzero proper restricted subalgebras.

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