A remark for those who, like me, feel a little rusty on the topic: even in characteristic $0$, a vanishing Killing for does not imply nilpotence of the Lie algebra (although the converse is true): the semidirect product of a one-dimensional algebra by an abelian ideal can provide a counterexample. (See these notes (“Introduction to Lie Algebras”) by N. Perrin, exercise 5.4.3, or this question.)
↧