Perfect Lie algebras

Question

How can I prove that $gl(n,k)$ and $sl(n,k)$ with $[x,y]=xy-yx$ are perfect algebras? By definition ,$g$ is a perfect algebra if $g=g\prime$, where $g\prime=<\{[x,y]| x,y\in g\}.$

Answer 1

Since $tr([A,B])=tr(AB)-tr(BA)=0$ it is clear that $[\mathfrak{gl}(n),\mathfrak{gl}(n)]\subseteq \mathfrak{sl}(n)$. The converse inclusion follows form a direct computation of commutators of basis matrices $E_{ij}$, having an entry $1$ at position $(i,j)$, and zero entries otherwise. So we have that $$ [\mathfrak{gl}(n),\mathfrak{gl}(n)]= \mathfrak{sl}(n), $$ so that $\mathfrak{gl}(n)$ is not perfect. The Lie algebra $\mathfrak{sl}(n)$ is simple (assuming that $k$ has characteristic zero). Any simple Lie algebra $L$ satisfies $[L,L]=L$, because $[L,L]$ is a nonzero ideal in $L$, so has to be $L$ itself.