# lie algebra semisimple?

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 $$[(X,v),(Y,u)] := ([X,Y],Xu-Yv).$$ This turns $L \times V$ into a perfect Lie algebra with $\text{Rad}(L \times V) = V$.