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 $z$?