If $\mathfrak g$ is a three dimensional Lie algebra and $[\mathfrak g,\mathfrak g]=\mathfrak g$. How to prove that there is a basis $\{x,y,z\}$ such that either
$[x,y]=z, [y,z]=x, [z,x]=y$ or
$[x,y]=2y, [x,z]=-2z,[y,z]=x$.
In case the field is $\mathbb C$, how to show that both lie algebras above are isomorphic?
Note: I only know basic theory of Lie algebras, so it would be great if the answers come with details..Thanks!
