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

  1. $[x,y]=z, [y,z]=x, [z,x]=y$ or

  2. $[x,y]=2y, [x,z]=-2z,[y,z]=x$.

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!


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