So I know the complex Lie algebra $sl(2,C)$ is isomorphic to the complexification of $su(2)$, $su_C(2)$. Since $\dim(su_C(2))=3$, then $sl(2,C)$ has also dimension 3. Therefore the complex Lie group $SL(2,C)$ (call this A) has also dimension 3. However, if we consider the decomplexification of $sl(2,C)$, or in other words, if we consider $sl(2,C)$ as a real vector space, its dimension is obviously 6. Then it seems there should be a corresponding Lie group $SL(2,C)$ (call this B) of dimension 6. I tried looking for such a Lie group, and some resources I found just state $SL(2,C)$ is of dimension 6 but doesn't mention its relation to (A).
My questions:
1) How can I distinguish by name the version of $SL(2,C)$ that is 3 dimensional (A) from the one that seems to exist as a 6 dimensional (B)? Are they called differently?
2) How are (A) and (B) related? How do you obtain one from the other? I mean it was easy to understand the Lie algebra of (B) is the decomplexification of the Lie algebra of (A), but it seems completely wrong to try to think of (B) being some sort of 'decomplexification' of (A). If you have any references you can suggest to understand better the version of $SL(2,C)$ that is 6-dimensional it would be really appreciated.
Thank you.