If L is a semisimple lie algebra then L=[L,L]. Is the opposite true?
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No. A Lie algebra satisfying that property is called perfect. For an example of a perfect Lie algebra that isn't semisimple, take a semisimple $L$ and an irreducible representation $V$ of $L$, and define a bracket on $L \times V$ by $$ [(X,v),(Y,u)] := ([X,Y],XuYv). $$ This turns $L \times V$ into a perfect Lie algebra with $\text{Rad}(L \times V) = V$. 

