If $L$ is a semi-simple Lie algebra, then $L=L'$.
Since $L$ is semi-simple we can write it as a direct sum of simple ideals $L_i$, i.e. $L=\oplus_{i=1}^r L_i$. Then $L'=\oplus_{i=1}^r L_i'$ and since every $L_i$ is simple we have $L_i'=L_i$ and consequently $$L'=\oplus_{i=1}^r L_i'=\oplus_{i=1}^r L_i=L$$
This proof seems surprisingly short. So my question is, did I make any mistakes or did I forget any crucial assumptions? Is there perhaps a more elementary proof?