Take the 2-minute tour ×  If L is a semisimple lie algebra then
L=[L,L]. Is the opposite true?  share|improve this question  1
Answer 1  up vote 18 down vote accepted  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
$\text{Rad}(L \times V) = V$.  share|improve this answer Maybe
it's also helpful to mention the substantial though often
isolated literature on perfect Lie algebras and related
structure theory? There are some interesting connections with
other questions, as in the paper by Benkart and Zelmanov in
Invent. Math. 126 (1996), 1-45. –  Jim Humphreys Apr 5 '11 at
13:08  Your Answer