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 $z$?



Restricted Lie algebras with no nonzero proper restricted subalgebras
For an element $a$ of $L$, denote by $\langle a \rangle\_p$ the restricted subalgebra generated by $a$.
Let $\mathbb{F}[t,\sigma]$ be the ring consisting of all polynomials $f=\sum\_{i\geq0}\alpha\_i t^{i}$ with respect to the usual sum and multiplication defined by the condition $t\cdot\alpha=\alpha^p t$ for every $\alpha \in \mathbb{F}$.


Then $L$ has no non-zero proper restricted subalgebras if and only if $L\cong \mathcal{L}/\langle \bar{f}\rangle\_p$, where $\mathcal{L}=\langle x\rangle\_p$ is a free cyclic restricted Lie algebra and $\bar{f}=\sum\_{i\geq0}\alpha\_ix^{[p]^i}$ is an element of $\mathcal{L}$ such that $f=\sum\_{i\geq0}\alpha\_it^{i}$ is an irreducible element of the ring $\mathbb{F}[t,\sigma]$.


The proof can be find here: <https://www.tandfonline.com/doi/abs/10.1080/03081087.2019.1708238?journalCode=glma20>


