I have just started learning about Lie algebras and got confused with this:
Some sources say the generators are $J_0,J_1$ and $J_2$ and some use $J_0,J_+$ and $J_-$. Which set is correct?
Or if both are correct what key concept am I missing here?
My understanding is that if have certain commutation relations then we know that Lie Algebra is such and such.
But if we have two such choices then this understanding falls apart?
How do I figure out what is $\mathfrak{su}(2)$ Lie Algebra in general?
Remember, there is the Lie group $SU(2)$ and its Lie algebra $\mathfrak{su}(2)$ which are two entirely different concepts. We are talking about the latter.
Anyway, a Lie algebra is in particular a vector space, and since $\mathfrak{su}(2)$ has dimension 3 (why?), it follows that the above matrix sets will generate it as a vector space, since they are linearly independent sets.
The Lie bracket is bilinear, so if we know what it does to a basis, we know the whole Lie algebra. This is why the commutation relations are practical to write down, but remember that it is basis dependent, so different generator sets will give different commutation relations.
Edit: Another notation thing. Don't use the word "representation" as in the title, representations of Lie algebras is a huge subject in itself. Presentation would be a better word, though I'm not sure if it is that fitting in the context of Lie algebras.