Let have the special linear algebra $\operatorname{Sl}(2,\mathbb F)$ ,which is the set of $ 2 \times 2$ matrix with trace zero. I have to prove that the lie algebra $ g=\operatorname{Span}\{ L_1,L_2,L_3\}=\mathbb CL_1+\mathbb CL_2+\mathbb CL_3$ where $ L_1,L_2,L_3$ are Angular Momentum, is the $ Sl(2,\mathbb C)$.
Contrariwise the set $ g=\operatorname{Span}\{ L_1,L_2,L_3\}=\mathbb RL_1+\mathbb RL_2+\mathbb RL_3$ is not the $\operatorname{Sl}(2,\mathbb R)$
I am trying to choose $h,x,y$ as linear combination of $ L_1,L_2,L_3$ such that: $[h,x]=2x , [h,y]=-2y , [x,y]=h$ , but i cant find the way .Can anyone help me? Thanks..