# Abelian simple Lie algebra has one dimension

In Humphreys' "Introduction to Lie algebras" it is told that we define a simple Lie algebra as non-abelian algebra because there is no need to pay much attention to one-dimensional case. Why abelian simple Lie algebra has to be a 1-dimensional one?

Because if $$\mathfrak g$$ is an abelian Lie algebra, then any vector subspace of $$\mathfrak g$$ is an ideal of $$\mathfrak g$$ and therefore $$\mathfrak g$$ is simple if and only $$\mathfrak g$$ is $$1$$-dimensional.
A simple Lie algebra is also semisimple. The fact that a finite-dimensional Lie algebra $$L$$ is semisimple over a field of characteristic zero is equivalent to the property that $$L$$ does not contain non-zero Abelian ideals. Hence a simple Lie algebra is never abelian. Therefore the $$1$$-dimensional Lie algebra is not simple.