The following table summarizes the properties of the various maps mentioned in the definition
When computing the root system for one of the simplest cases of Lie Groups, the group SL(2, R) of two dimensional matrices with determinant 1 consists of the set of matrices of the form:
A maximal compact connected abelian Lie subgroup, or maximal torus T, is given by the subset of all matrices of the form
If we conjugate an element of SL(2, R) by an element of the maximal torus we obtain
It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be proved that the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, R).