Killing form for a non-abelian Lie Algebra of dimension $2$

Question

Aratati ca forma Killing pentru algebra Lie ne-abeliana de dimensiune 2 nu este zero.

How can I prove that the Killing forme of a non-abelian Lie algebra is not equal with $0$?

Thanks

Answer 1

That is not true. For example, the Killing form of any nilpotent algebra is zero. In some books (for example in my version of Kirillov Jr. book) it is stated, that the converse is true, but that is wrong.