The main point of the question is: I would like to know whether there are only finitely many, countable infinitely many or even uncountable many isomorphism classes of 3dimensional real lie algebras...

For 2 dimensions, I know that there are exactly 2 real lie algebras but for 3 dimensions I could not find any reference in the literature...

And even more interesting: Let U be an open ball in 3dimensional Euclidean space. Then U is a 3dimensional manifold. How many non-isomorphic Lie-Group strucutres exist on U?
This question is of course related to the Lie algebra question because every Lie group structure on U gives rise to a Lie algebra, but we dont get all Lie algebras since there are 3dimensional real Lie algebras whose simply connected real Lie group is compact and hence not diffeomorphic to U (for example su(2) )

I would be very grateful, if someone could help me out here.

Thanks.