What is good about simple Lie algebras? Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple.
I wonder what beautiful properties does a simple Lie algebra have? 
Well, I've only learned linear algebra, mathematical analysis and a bit of abstract algebra, so I might expect a more amateur answer~
I think there may be other readers of the book sharing the same question. And a good answer may help us gain more motivation to learn about Lie theory~
Much thanks!!!
 A: First of all, we have a complete classification of the (complex) simple Lie algebras. Namely, we have that every simple Lie algebra is one of the following:


*

*$\mathfrak{sl}(n)$

*$\mathfrak{so}(n)$

*$\mathfrak{sp}(2n)$

*one of the five exceptional simple Lie algebras $\mathfrak{e}_6$, $\mathfrak{e}_7$, $\mathfrak{e}_8$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$


This gives us directly a classification of all semisimple Lie algebras (i.e. the "well behaved" Lie algebras). It also helps to study general Lie algebras through Levi's theorem, which says that every Lie algebra is the semi-direct product of its radical (i.e. its maximal solvable ideal) and a simple Lie algebra.
A: Simple, and more generally semisimple Lie algebras are very nice because of their structural properties and that they can be classified in an elegant way. 
The significance of semisimplicity also comes from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. 
