No Lie algebra over $\Bbb R$ or $\Bbb C$ can have a unit element.

Question

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.

Answer 1

It would also contradict Jacobi's identity. If your Lie algebra is commutative, then clearly you can't have that $[x,e]=x$ unless $x=0$ since $[x,e]$ would have to be $0$. If it is non-commutative (meaning that for some $x,y$ in the Lie algebra we have that $[x,y]\neq 0$), then Jacobi's identity would say that

$$0 = [x,[y,e]] + [y,[e,x]] + [e,[x,y]] = [x,y] + [y,x] + [x,y] = [x,y]$$

by anti-symmetry. This holds for all $x,y$ in the Lie algebra which is a contradiction.

Answer 2

If $[x,e]=[e,x]$, then what does the antisymmetry of the Lie bracket tell you?

Answer 3

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