Theorem- Up to isomorphism, the only noncommutative Lie algebra of dimension 2 is that with basis $x , y$ and bracket determined by $[x,y] = x$.
I understand that all vector spaces of dimension 2 over the field $K$ are isomorphic to each other. So the number of lie algebras of dimension 2 in a field $K$ is determined by the number of possible bilinear operations [ ]$:\ V \ X \ V \rightarrow V$ satisfying the conditions
$a)$ $[x,x]=0$ for all $x\in V$
$b)$ $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z \in V$
The bilinear operations on the other hand is determined by the elements to which the pair of base elements are mapped to in the bilinear operation. And since in a lie algebra $[x,x]=[y,y]=0$ and $[x,y]=-[x,y]$ we ony need to determine $[x,y]$. Now how do we prove that $[x,y]=x$ and $[y,x]=-x$ always and why can't it be [y,x]=y or any other vector ?