TITLE: Lie algebras with abelian Cartan subalgebras
QUESTION [12 upvotes]: 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.

REPLY [2 votes]: If $H$ is a Cartan subalgebra of a Lie algebra $L$, and $A$ is an associative commutative algebra, then $H \otimes A$ is a Cartan subalgebra of $L \otimes A$. Specializing this to the case $L$ classical simple, we get another positive answer to the first question (but not to the second one). This example can be, probably, varied, by taking subalgebras of $L \otimes A$, or adding "tails" of derivations, etc.