Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.


I wonder what is known/expected for char p=2,3 ?


More vague and soft question is the following - look at some famous classification problems: simple finite-dim Lie algebras, simple finite groups, some other things classified by ADE...
We see the following pattern: there are some series of objects and finite number of "sporadic" objects. I.e. it never happens that there is infinite number of examples which are not in "series". 
So classification of simple objects is simple (in some very informal sense).


The question: can we expect this in advance, without obtaining classification ? (What are other examples/counter examples of similar phenomenon ?).


For example can we expect/prove this for simple Lie algs for char =2,3 ? I.e. there will be some finite number of series and finite number of "sporadic" examples ?



Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...  
According with the introduction of Strade's book "Simple Lie algebras over fields of positive characteristic. Structure Theory", it seems that a possible list of known finite-dimensional simple Lie algebras over algebraically closed fields of characteristic 3 could be close to complete. A discussion on this topics can be found in Section 4.4 of that book. On the other hand, in characteristic 2 the situation seems to be more complicated. For example, in the paper [Yu. Kochetov - D. Leites: Simple Lie algebras of characteristic 2 recovered from superalgebras and on the notion of a simple group, in Proceedings of the International Algebraic Conference in the Memory of A.I. Malcev, Novosibirsk, Contemp. Math. 131 (1992), 59-67] the authors have constructed simple Lie algebras in characteristic 2 from superalgebras. Thus one expects that a greater variety of constructions could get many more examples in this exceptional characteristic. 


