[Math] How many three dimensional real Lie algebras are there

Question

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.

Answer 1

There are already uncountably many isomorphism classes of $3$-dimensional real Lie algebras. In fact, there are $1$-parameter families of $3$-dimensional solvable Lie algebras. The classification has been done over an arbitrary field. For references see the paper of Willem A. de Graaf, and the book of Jacobson on Lie algebras, chapter $1$, section $4$.