Cartan subgroup

A cartan subgroup is like a special type of group that helps us understand other groups. Think of it like a group of friends, but these friends are special because they help us understand other groups of people.

In math, we talk about something called a "Lie group," which is basically a mathematical group that describes the symmetries of some kind of object or system. (Don't worry if you don't understand what that means, just think of it like a group of friends who all share something in common.)

Now, one thing we can do with Lie groups is break them down into smaller, simpler groups. This is where the cartan subgroup comes in. Think of it like taking a big group of friends and breaking them up into smaller groups that all have something in common.

The cartan subgroup is special because it helps us break down a Lie group into even smaller groups, until we get to the simplest possible groups (which are called "cartan subalgebras").

So, think of it like we have a big group of friends, and we want to understand them better by breaking them up into smaller groups that all have something in common. The cartan subgroup helps us do this, and eventually we end up with the simplest possible groups that help us understand this big group of friends better.