Abstract: The classification of simple Lie algebras over the
reals and complexes has been known for a long time. But over
an arbitrary field, the question is quite a bit more complicated.
(For example, in characteristic $p$, one should probably look
at algebraic groups instead of Lie algebras.) The classification
is still very far from being resolved, and some conjectures due
to Serre and others are equivalent to special cases of it.