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 $3$-dimensional 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 $3$-dimensional 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 $3$-dimensional real Lie algebras whose simply connected real Lie group is compact and hence not diffeomorphic to $U$ (for example $\frak su (2)$ )

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

Thanks.

only$3$-dimensional Lie algebra whose corresponding connected, simply-connected Lie group is not diffeomorphic to $\mathbb{R}^3$. – Robert Bryant May 9 at 15:38