This Week's Finds 6: Coxeter and Dynkin diagrams

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. Here I use Dynkin diagrams to classify compact Lie groups - and especially compact semisimple Lie groups.
I made 3 mistakes. First, while it's true that a compact Lie algebra is 'semisimple' iff for every z there exist x,y such that [x,y] = z, and I was mainly talking about compact Lie algebras, this is not a valid definition of semisimplicity for arbitrary Lie algebras. A better definition of semisimplicity is to require the 'Killing form' to be nondegenerate:
en.wikipedia.org/wiki/Killing...
The Killing form for is something you can define for any Lie algebra, and for a compact semisimple Lie algebra, the negative of the Killing form is what I called the 'god-given inner product'.
Second, the Weyl group is not the normalizer of the maximal torus; it's the normalizer mod the centralizer. I fixed this during the talk:
en.wikipedia.org/wiki/Weyl_gr...
Third, the root lattice associated to the Lie algebra of a compact semisimple Lie group G is actually the dual lattice of the kernel of the exponential map exp: Lie(T) → T, where T is the maximal torus - and even this is only true when G has trivial center. But every compact semisimple Lie group G is a covering space of a unique compact semisimple Lie group with trivial center. What I called the root lattice is actually what Adams calls the 'integer lattice', and I recommend his treatment of these issues:
J. F. Adams, Lectures on Lie Groups, University Of Chicago Press, 1983.
For more, read my paper "Coxeter and Dynkin diagrams" here:
math.ucr.edu/home/baez/twf/
This is one of a series of lectures at the University of Edinburgh on topics drawn from my column This Week's Finds.
Cover image by Tom Ruen, CC BY-SA 4.0