Bianchi classification

Remarks

The proof follows the dimension of the derived algebra $L' = [L, L]$. If $L' = L$, then $L$ is simple, and its complexification is $\mathfrak{sl}_2(\C)$; the two real forms are $\mathfrak{sl}_2(\R)$ and $\mathfrak{so}_3(\R)$. The Killing form is indefinite on the first and negative definite on the second, so they are not isomorphic. Otherwise $L$ is solvable and has a $2$-dimensional abelian ideal, so it is a semidirect product $\R \ltimes_A \R^2$. Replacing $x$ by $cx + v$ with $c \ne 0$ and $v \in \R^2$ replaces $A$ by $cA$, and a change of basis of $\R^2$ conjugates $A$. So the type depends only on $A$ up to conjugation and nonzero scaling, which is what the list enumerates. Types I, II, $\mathrm{VI}_0$, $\mathrm{VII}_0$, VIII and IX are the unimodular ones, where every $\operatorname{ad} y$ has trace zero.

Going deeper

For a student

The list is the real counterpart of the classification of $3$-dimensional complex Lie algebras. Over $\R$, a $2 \times 2$ real matrix can have complex eigenvalues, which gives the extra family $\mathrm{VII}_h$, and $\mathfrak{sl}_2(\C)$ has two real forms. In general relativity, the Bianchi types classify spatially homogeneous cosmological models by the Lie algebra of their $3$-dimensional isometry group acting simply transitively. The unimodular types form class A and the others class B.