TITLE: perfect Lie algebra but not semisimple
QUESTION [6 upvotes]: Lie algebra $\mathfrak{g}$ is perfect if and only if $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$. 
And we know semisimple Lie algebra must satisfy the $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$ then it's perfect.
What is the example of perfect Lie algebra but not semisimple? And what's the sufficient and necessary condition for a perfect Lie algebra to be semisimple Lie algebra?

REPLY [2 votes]: Take any semisimple Lie algebra $L$ of dimension $n$ and an irreducible representation $V$ of $L$, of dimension $m \ge 2$ 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 $\text{Rad}(L \times V) = V$. The dimension is $n+m$. It is not semisimple.
A necessary and sufficient condition for a perfect Lie algebra to be semisimple is that its solvable radical is trivial.