Abelian simple Lie algebra has one dimension

Question

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?

Answer 1

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.

Answer 2

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.