[en] The Cartan-Dynkin theory of classical groups is considered with the aid of a lemma proposed in this work. The lemma says that all the Weyl conjugates of a dominant weight will have a common form with the dominant weight when they are expressed as linear combinations of fundamental representation weights (FRW) of the group. This reformulation of the Cartan-Dynkin theory simplifies the calculations concerning grand unifying schemes. Especially, the specification of the weights of a representation, the property of being real or complex, the branchings of a representation under its subgroups and the decomposition of a direct product are reduced to simple permutational calculations. We introduce the notion of ''modular decomposition'' to study the group representations. This work includes only the groups SU(N+1) because they have characteristics different from others, such as SO(2N+1), SP(2N), SO(2N) and exceptional groups. In the two successive applications the idea shall be completed also for these groups. (author)