Given an 8 dimensional Lie algebra with generators A B C D E F G H. If it is the case that A B C D all commute with each other, and if you add any of E F G H, you end up breaking commutativity, then the collection A B C D is called a Cartan subalgebra and it has rank 4. For a given Lie group, even this one, it can be proven that all Cartan subalgebras have the same rank. For example, another Cartan subalgebra could very well be E F G H, as long as these 4 commute and adding any of A B C D would break commutativity.
You can say this: The Cartan subalgebra picks out the largest set of directions where flows don’t interfere with each other.
