Weights, Roots and Coweights of a Lie group

Suppose you have a Lie group. The generators of this group, along with their commutation relations, form a Lie algebra. The maximal commuting subset of this is called a Cartan Sub-Algebra (CSA). Suppose you have a CSA with 20 generators. Given some vector space V, each of the 20 generators can be associated with 20 matrices. If you change V, you change the 20 matrices If a vector A of the vector space V is an eigenvector of every element of the CSA (let’s call them M1, M2, M3 … M19, M20), then M