Classification of low-dimensional Lie algebras

Remarks

In dimension $2$, if $L$ is not abelian then $L'$ is a line $k y$; choosing $x$ with $[x, y] \ne 0$ and rescaling gives $[x, y] = y$. In dimension $3$ the proof splits by $\dim L'$. If $\dim L' = 2$, one shows that $L'$ is abelian and that $\operatorname{ad} x$ acts on $L'$ by an invertible map for $x \notin L'$. Its Jordan form, up to scaling $x$, gives cases 4 and 5. If $L' = L$, one shows that some $\operatorname{ad} h$ has a nonzero eigenvalue, and its eigenvectors give an $\mathfrak{sl}_2$-triple that spans $L$. Over $\R$ the list is longer, since $\operatorname{ad} x$ may have non-real eigenvalues and $\mathfrak{sl}_2(\R)$ and $\mathfrak{so}_3(\R)$ are not isomorphic; the real case is the Bianchi classification.

Going deeper

For a student

The invariant that organizes the proof is the derived algebra $L' = [L, L]$ and its relation to the center. The solvable cases, with $L' \ne L$, reduce to linear algebra: a $2$-dimensional abelian ideal and one operator acting on it. The case $L' = L$ forces $L \cong \mathfrak{sl}_2$, the smallest simple Lie algebra. In higher dimensions the lists grow quickly and contain families with several parameters.