I found the following exercise at page 85 of the Strade-Farnsteiner's book "Modular Lie algebras and their representation": Let $D$ be a finite-dimensional division ring over a field $F$ of characteristic $p>0$. Consider $D$ as a restricted Lie algebra via the Lie product $[x,y]=xy-yx$ and $p$-mapping $x \mapsto x^p$. Show that every element $x \in D$ is semisimple. Is this claim actually true in general?
The claim is true under the assumption that the ground field $F$ is perfect. This follows from Exercise 2 in the same page which is, in turn, an immediate consequence of Theorem 3.7 in the same section of the book of Strade and Farnsteiner. On the other hand, if $F$ is not perfect then the previous result does not hold in general, as an inseparable extension field of $F$ already shows.