Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

Question

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then $[L,Rad(L)] \subseteq N$. It's obvious that $[L,Rad(L)]$ is a solvable ideal in $L$, but I can't see how to prove its nilpotency. I found a proof in Bourbaki's book, but it's rather long and uses representation theory. Is there an easier way to prove it?