Did the Ante-Nicene Fathers Teach or Believe in the Trinity?

Some of the most significant extant religious documents available to us
at present are the writings of the ante-Nicene Fathers. The voluminous
literary corpus produced by the early Church Fathers is crucial since
it provides a glimpse into the early Church's unique belief system. In
short, the ante-Nicene library of works contains early Christianity's
doctrine of God. Most important for our present purposes, these
theological documents help us to appreciate the early Church's position
on the Trinity and its view of Christ in God's divine purpose. With
regard to the Trinity doctrine, however, what was the view of the
ante-Nicene ecclesia? Did the ante-Nicene fathers teach that God is
threefold? Alternatively, did these men espouse a subordinationist
view?

When reading the last question, some readers may feel that we are
guilty of the either/or (vel/vel) fallacy. "Stop question loading!"
logicians may assert. Yes, certain readers may think that it is
erroneous to assume that the ante-Nicene fathers were either
Trinitarians or subordinationists. Why could they not possess both
viewpoints simultaneously?

Once a proper understanding of subordinationism is grasped, it will
become clear that there is no possible way a Christian can
simultaneously affirm both subordinationism and Trinitarianism. Why is
this the case though?

First, we think that the very definition of subordinationism makes it
logically impossible to concomitantly affirm Trinitarianism and
subordinationism. But to fully understand this particular contention,
it is imperative to define the subordinationist position, restate the
claims of classical Trinitarianism, and then juxtapose those claims
with the ante-Nicene writings.This study will endeavor to successfully
navigate through the torturous but exciting field of arcane Trinitarian
terminology and clearly show the necessarily contrasting positions of
subordinationists and Trinitarians. We will conclude that the
ante-Nicene fathers were not Trinitarians qua Trinitarians.

Defining the Trinity Doctrine

As noted by contemporary theologians, there are in fact many disparate
"doctrines" of the Trinity (i.e., the Trinity doctrine is not
monolithic). For instance, systematic theologian Owen Thomas observes:

Our survey of the history of the [Trinity] doctrine in the text has
indicated that there are several doctrines of the trinity: Eastern,
Western, social analogy, modal, so forth. There is one doctrine in the
sense of the threefold name of God of the rule of faith as found, for
example, in the Apostle's Creed. This, however, is not yet a doctrine.
It is ambiguous and can be interpreted in a number of ways. There is
one doctrine in the sense of the Western formula of "three persons in
one substa