How can I prove that a two-dimensional Lie algebra cannot be simple

Question

I am currently studying Lie algebra and now I am confused about how could I prove that a two-dimensional Lie algebra cannot be simple? Thanks for any answers in advance.

Answer 1

If $L$ is two-dimensional either $[L,L]$ is zero-dimensional, in which case $L$ is Abelian, or it is one-dimensional, in which case it's an ideal of $L$.