Lie's theorem in characteristic $p$

Lie’s theorem indeed still holds in positive characteristic provided the dimension of the vector space is less than the characteristic. For reference see, for example, the remark before example $81$ in http://math.berkeley.edu/~reb/courses/261/11.pdf, It is indeed often the case that results being true in characteristic zero, remain true for $p$ larger than the dimension of the vector space (or larger than the Coxeter number for simple Lie algebras). On the other hand there may be many results from the modular world where we need to be careful with such a statement. The classification of simple Lie algebras from characteristic zero does not remain true for large $p$.

This is addressed in G. Seligman's Modular Lie Algebras, chapter V §1, with more references therein. In particular Seligman writes "some of the proofs referred to above [among them is Jacobson's in Lie Algebras, pp. 43--50] are still applicable when the degree of the matrices is less than the characteristic". He also discusses (non-)validity of some corollaries of Lie's theorem in prime characteristic.