Lie Algebras with lower dimension

Question

$\forall\ L$ is a Lie algebra, if $\rm{dim}_{\mathbb{R}}L=2$, can we have $L$ is solvable? Is there any example of $L$ which fits these condition but $L$ is not nilpotent?

Answer 1

There are exactly two different Lie algebras over a field $K$, namely the abelian Lie algebra $K^2$ and the non-abelian Lie algebra $\mathfrak{r}_2(K)$, with basis $(x,y)$ and Lie bracket $[x,y]=x$. Both Lie algebras are solvable. The non-abelian Lie algebra has trivial center, hence cannot be nilpotent.

References:

Two Dimensional Lie Algebra

Two dimensional Lie Algebra - what do we know without knowing the Bracket?