0

Let $L$ be a real Lie algebra. I want to show that $Z(L_\mathbb{C})=Z(L)_\mathbb{C}$. I already proved that $Z(L)_\mathbb{C}\subseteq Z(L_\mathbb{C})$. Can you give me a hint how I can show the other direction?

José Carlos Santos
  • 427,504
user95
  • 89

1 Answers1

1

Write $x=u+iv\in L_{\Bbb C}$ with $u$, $v\in L$. If $x$ is in the centre of $L_\Bbb C$ then certainly $[x,y]=0$ for all $y\in L$. As $[x,y]=[u,y]+i[v,y]$ then $[u,y]=[v,y]=0$ so that $u$, $v\in Z(L)$.

Angina Seng
  • 158,341