TITLE: No Lie algebra over $\Bbb R$ or $\Bbb C$ can have a unit element.
QUESTION [6 upvotes]: I want to prove: No Lie algebra over $\Bbb R$ or $\Bbb C$ can have a unit element.

Now I am not sure how to take this in regard to the Lie bracket. I.e. I have now idea where to start. $[x,e]=[e,x]=x$ or something? That doesn't make sense to me, since we are just looking at an abstract bracket it seems.

REPLY [3 votes]: If $[e,x]=x$ for all $x$, what can you conclude from $0=[e,e]$?