  For any Carter diagram $\Gamma$ containing 4-cycle, we introduce the partial
Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a
Dynkin diagram. A linkage diagram is obtained from $\Gamma$ by adding one root
together with its bonds such that the resulting subset of roots is linearly
independent. The linkage diagrams connected under the action of dual partial
Weyl group (associated with $B_L$) constitute the linkage system, which is
similar to the weight system arising in the representation theory of the
semisimple Lie algebras. For Carter diagrams $E_6(a_i)$ and $E_6$ (resp.
$E_7(a_i)$ and $E_7$; resp. $D_n(a_i)$ and $D_n$), the linkage system has,
respectively, 2, 1, 1 components, each of which contains, respectively, 27, 56,
$2n$ elements. Numbers 27, 56 and $2n$ are well-known dimensions of the
smallest fundamental representations of semisimple Lie algebras, respectively,
for $E_6$, $E_7$ and $D_n$. The 8-cell "spindle-like" linkage subsystems called
loctets play the essential role in describing the linkage systems. It turns
that weight systems also can be described by means of loctets.
