  A diagram obtained from the Carter diagram $\Gamma$ by adding one root
together with its bonds such that the resulting subset of roots is linearly
independent is said to be the {\it linkage diagram}. Given a linkage diagram,
we associate the linkage labels vector, which is introduced like the vector of
Dynkin labels. Similarly to the dual Weyl group, we introduce the group
$W^{\vee}_L$ associated with $\Gamma$, and we call it the dual partial Weyl
group. The linkage labels vectors connected under the action of $W^{\vee}_L$
constitute the linkage system $\mathscr{L}(\Gamma)$, which is similar to the
weight system arising in the representation theory of the semisimple Lie
algebras. The Carter theorem states that every element of a Weyl group $W$ is
expressible as the product of two involutions. We give the proof of this
theorem based on the description of the linkage system $\mathscr{L}(\Gamma)$
and semi-Coxeter orbits of linkage labels vectors for any Carter diagram
$\Gamma$. The main idea of the proof is based on the fact that, with a few
exceptions, in each semi-Coxeter orbit there is a special linkage diagram --
called {\it unicolored}, for which the decomposition into the product of two
involutions is trivial.
