I assumed that the 1-dimensional Lie algebra was simple since I cannot think of any proper non-trivial ideal it could have (you either have no elements, or once you have one element you span the space). However every classification of simple Lie algebras I have seen does not include a 1-dimensional Lie algebras. Is there a reason for this or is the 1-dimensional Lie algebra somehow simple?
Thank you!
A Lie algebra is said to be simple if it is not Abelian and has no nonzero proper ideals. So we want to exclude the $1$-dimensional Lie algebra from the simple Lie algebras. In dimension $2$ there are only two non-isomorphic Lie algebras over any field, the abelian one and the non-abelian solvable one. So the first simple Lie algebra we have has already dimension $3$, for example $\mathbb{sl}_2(K)$ for a field of characteristic not $2$.