The Cartan subalgebras of a reductive Lie algebra are abelian. 

> Are there non-reductive Lie algebras with abelian Cartan subalgebras?

In fact, the elements of a Cartan subalgebra of a reductive Lie algebra are semisimple, so a weaker question is:

> Are there non-reductive Lie algebras with abelian Cartan subalgebras all of whose elements are semisimple?

**N.B.:** I asked this earlier at [math.SE](https://math.stackexchange.com/q/110294/274).