Cohomology
For a non-abelian connected compact Lie group G, the first group cohomology of the Weyl group W with coefficients in the maximal torus T used to define it, is related to the outer automorphism group of the normalizer as:
The outer automorphisms of the group Out(G) are essentially the diagram automorphisms of the Dynkin diagram, while the group cohomology is computed in (Hämmerli, Matthey & Suter 2004) and is a finite elementary abelian 2-group ; for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.
Read more about this topic: Weyl Group