= Classification of three-dimensional perfect Lie algebras
{title2=$\dim\mathfrak g=3,\ [\mathfrak g,\mathfrak g]=\mathfrak g\ \Longrightarrow\ \mathfrak g\cong\mathfrak{sl}_2(k)$}
Over an <algebraically closed field> of <characteristic zero>, every three-dimensional <perfect Lie algebra> is isomorphic to the <sl2 Lie algebra>. Indeed the <trace of a commutator> gives $\operatorname{tr}(\operatorname{ad}x)=0$ for all $x$, because each element is a sum of <Lie brackets>. If every $\operatorname{ad}x$ were a <nilpotent linear map>, the <Engel theorem> would make the <Lie algebra> nilpotent, contradicting its constant nonzero <lower central series of a Lie algebra>. Some $\operatorname{ad}x$ therefore has a nonzero <eigenvalue> $\lambda$. Its other <eigenvalues> are $0,-\lambda$, since it kills $x$ and has zero <trace>. They are distinct, so its <eigenvectors> give a <basis> $h,e,f$ after scaling $x$ to $h=2x/\lambda$, with $[h,e]=2e$ and $[h,f]=-2f$. The <Jacobi identity> puts $[e,f]$ in the zero <eigenspace> $kh$. Its coefficient cannot vanish, since otherwise the <derived algebra> would lie in $ke\oplus kf$. Rescaling $f$ gives $[e,f]=h$. Thus this <basis> is an <sl2 triple>, proving the claimed <Lie algebra isomorphism>.
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