  In 1972, R. Carter introduced admissible diagrams to classify conjugacy
classes in a finite Weyl group W. We say that an admissible diagram \Gamma is a
Carter diagram if any edge {\alpha, \beta} with inner product (\alpha, \beta) >
0 (resp. (\alpha, \beta) < 0) is drawn as dotted (resp. solid) edge. We
construct an explicit transformation of any Carter diagram containing long
cycles (with the number of vertices l > 4) into another Carter diagram
containing only 4-cycles. Thus, all Carter diagrams containing long cycles can
be eliminated from the classification list. There exist diagrams determining
two conjugacy classes in W.It is shown that any connected Carter diagram \Gamma
containing a 4-vertex pattern D_4 or D_4(a_1) determines a single conjugacy
class. The main approach is studying different extensions of Carter diagrams.
Let \tilde{\Gamma} be the Carter diagram obtained from a certain Carter diagram
\Gamma by adding a single vertex \alpha connected to \Gamma at n points, n \leq
3. Let a socket be the set of vertices of \Gamma connected to \alpha. If the
number of sockets available for extensions is equal to 2, then there is a pair
of extensions \Gamma < \tilde{\Gamma}_L and \Gamma < \tilde{\Gamma}_R, called
mirror extensions and the pair elements w_L and w_R associated with
\tilde{\Gamma}_L and \tilde{\Gamma}_R. We show that w_R = T^{-1}w_L{T} for some
T \in W, where the map T is explicitly constructed for all mirror extensions.
In Carter's description of the conjugacy classes in a Weyl group a key result
(Carter's theorem) states that every element in a Weyl group is a product of
two involutions. One of the goals of this paper and its sequels is to prepare
the notions and framework in which we give the proof of this fact without
appealing to the classification of conjugacy classes.
