Existence of Lie algebra with dim =3 or ≥ 5 with [g,g]=g

Question

How to show that: There is a Lie algebra $\mathfrak g$ of dimension $k = 3$ or $k≥ 5$ iff $\frak g=[g,g]$. Also, why it is possible to choose $\frak g$ such that its center is $0$. However, for the cases where $k = 5$ or $k= 7$ then $\frak g$ can't be semisimple?

Answer 1

A Lie algebra $L$ satisfying $L=[L,L]$ is called perfect. Every semisimple Lie algebra is perfect, so $\mathfrak{sl}(2)$ is a perfect Lie algebra of dimension $k=3$.

For an example of a perfect Lie algebra that isn't semisimple, take a semisimple $L$ and a nontrivial 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 $\text{Rad}(L \ltimes V) = V$ and center $0$. For example, take $\mathfrak{sl}(2)\ltimes V(m)$ of dimension $m+3\ge 5$, to obtain the claim.