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$3$-dimensional real lie algebras...
For 2$2$ dimensions, I know that there are exactly 2$2$ real lie algebras but for 3 dimensions I could not find any reference in the literature...
And even more interesting: Let U$U$ be an open ball in 3dimensional$3$-dimensional Euclidean space. Then U$U$ is a 3dimensional manifold. How many non-isomorphic Lie-Group strucutres exist on U$U$? This question is of course related to the Lie algebra question because every Lie group structure on U$U$ gives rise to a Lie algebra, but we dont get all Lie algebras since there are 3dimensional$3$-dimensional real Lie algebras whose simply connected real Lie group is compact and hence not diffeomorphic to U $U$ (for example su(2)$\frak su (2)$ )
I would be very grateful, if someone could help me out here.
Thanks.