I think a few people are making this harder than it needs to be.
Let’s put aside potential subtleties about elementary vs. individual vs. infinitesimal relatives and simply use “elementary relative” to cover all cases at a first approximation. One of the advantages of this approach is the analogy it highlights between elementary relations in the logic of relatives and elementary transformations in linear algebra, affording a bridge to practical applications of relation theory.
The time has come for a concrete example. Suppose we have a universe of discourse X consisting of biblical figures.
Linguistic phrases like “brother of __” or “x is y’s brother” and many others may be used to indicate a dyadic relation B forming a subset of X × X such that (x, y) is in B if and only if x is a brother of y.
It is often convenient to use Peirce’s notation x:y for the ordered pair (x, y). Among other things it’s easier to type on the phone.
In the universe X of biblical figures, Cain:Abel is an elementary relation in the brotherhood relation B.
But Cain:Abel also belongs to the relation E indicated by “elder brother of” and again to the relation S indicated by “slayer of”. So the elementary relation by itself does not completely determine the general relation or general relative term under which it may be considered.
This means that classifying relations is a task at a categorically higher level than classifying elementary relations.
In the special case of triadic sign relations, almost all the literature so far has tackled only the case of elementary sign relations.