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.