$\forall\ L$ 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?
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Two dimension is easy to calculate by hand. Can you even write up a nontrivial Lie bracket on a 2d real vector space? – Berci Dec 23 '19 at 20:32
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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 - what do we know without knowing the Bracket?
Dietrich Burde
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