TITLE: Restricted Lie algebras with no nonzero proper restricted subalgebras
QUESTION [6 upvotes]: 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 $z$?

REPLY [2 votes]: 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