Cartan decomposition

Imagine you have a box with various colored blocks inside. Now, you can take these blocks apart and divide them into two groups - those that are square and those that aren't square.

In the same way, when we talk about a Cartan decomposition, we're talking about how we can divide a group, which is a collection of mathematical operations or transformations, into two different types of parts, just like we separated the blocks.

One part is called a Cartan subgroup, and this is like the square blocks - they don't change the overall shape of the group in any meaningful way. You can think of this subgroup as a sort of foundation or building block for the group.

The other part is called the Cartan complement, and this is like the non-square blocks. They add more detail and variety to the group, but they're not essential to its basic structure.

Now, the reason this is useful is that it helps us understand more about the structure of the group we're working with. Just like we could look at the contents of our box and say "there are more square blocks than non-square blocks," we can look at the Cartan decomposition and say things like "the Cartan subgroup is a compact, connected Lie group," or "the Cartan complement is isomorphic to a vector space."

So, a Cartan decomposition is just a way of breaking down a complicated mathematical object into its essential building blocks and additional details, so we can better understand how it works.