The following question is from Erdmann and Wildon Introduction to Lie algebras.( Chapter 1 exercises) I am a teacher of mathematics in a school and I study advance mathematics as a hobby.
Question: Let $L$ be a 3 -dimensional complex Lie algebra with basis $(x,y,z)$ and Lie bracket defined by $[x,y]=z , [y,z]=x,[z,x]=y$ . Find an explicit isomorphism between $sl(2,\mathbb{C})$ and $L$.
Here $sl(2,\mathbb{C})$ is set of all $2\times 2$ matrices with trace$=0$ over $\mathbb{C}$.
Let $\phi$ denote the isomorphism that I must construct. The $x,y, z $ which are basis elements must go to $e,f,g$ basis elements of $sl(2,\mathbb{C})$.
But how to construct such a map ? Can you please help me with that?
After that I need to show it is a homomorphism, one -one , onto.