We start with an easy exercise.
ExerciseLet be the -dimensional complex Lie algebra with basis such that Find a Cartan decomposition of .
Note that this Lie algebra is not nilpotent, because is not ad-nilpotent: It is clear that is a nilpotent Lie subalgebra of of dimension , and since is not nilpotent, we see that is a maximal nilpotent subalgebra of . In fact, is a maximal abelian subalgebra of . However, is not a Cartan subalgebra. For example, note that is not self-normalized, because but .
A Cartan subalgebra of can be chosen, for example, , as it is easy to see that is self-normalized, and is of course nilpotent. It turns out that and are both maximal nilpotent subalgebra, and has smaller dimension than .
A Cartan decomposition of is
Although is not a Cartan subalgebra, we can still find the weight space decomposition of as acts on . Since and are ad-nilpotent, the weight space decomposition is just Note that this generalized weight space (with respect to weight ) is strictly larger than , which is not strange because is not a Cartan subalgebra.