The isomorphism type of the centralizer of an element in a Lie group

Let G be a compact 1-connected simple Lie group, and let x∈G be a group element. We determine the isomorphism type of the centralizer Cx in term of a minimal geodesic joining the group unit e∈G to x.This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G (Borel and De Siebenthal, 1949 [4]), and the isomorphism types of parabolic subgroups of G.