Why does a simple Lie algebra have zero radical?

Question

I know a simple Lie algebra $L$ has only two ideals: $\lbrace0\rbrace$ and itself. If $\lbrace0\rbrace$ is the radical (i.e., maximal solvable ideal), we are done. If $L$ is the radical... where is the contradiction?

I'm sure this is really simple.

Answer 1

The solvable radical $rad(L)$ is an ideal of $L$. So it must be $0$ or $L$, for $L$ being simple. If $rad(L)=L$, then $L$ is simple and solvable at the same time, hence $0$. If $L\neq 0$, then consequently $rad(L)=0$.