Classification of simple (or simple-restricted) Lie algebras over algebraically closed fields in positive characteristic is studied for a long time. Today, we know all finite-dimensional simple (or simple-restricted) Lie algebras over algebraically closed fields of characteristic $p \ge 5$. But, how about over finite fields? I don't know either the results of classifications over finite fields or some examples. Does anyone know some references?
Lie Algebras – Classification of Simple Lie Algebras over Finite Fields
finite-fieldslie-algebraspositive-characteristicrestricted-lie-algebras
Best Answer
The classification of simple finite-dimensional Lie algebras over finite fields is a very hard task and only few results are known. However, I suggest to have a look at the following paper by Bettina Eick and references therein: Some new simple Lie algebras in characteristic 2.