This is a bit of an odd question perhaps but I found myself asking the "why" question when studying "Lie Algebras in particle physics" by Howard Georgi concerning the previously mentioned objects.
Why do we introduce and study simple roots, Dynkin diagrams and Cartan matrices? What I gather from the book is that they offer a way of "compressing" the "information" of a Lie Algebra immensely in the sense that all the properties of a Lie Algebra can be encoded within Dynkin diagrams/simple roots and then the entire root system and the commutation relations of the algebra can then be reconstructed using the Cartan matrix and other tools. But of what use is all that?
It's not like one couldn't just draw the whole root systems or give the commutation relations of the Lie Algebra when one wants to communicate the structure of an algebra, right?