No (good) trinitarian theologian wants to have a fourth thing - a divine substance considered apart from the Three Persons. But it's important to be aware that this error (effectively having a quaternity) has two versions. There is a vulgar quaternity and a more insidious one.
The vulgar one looks like this:
Here is the "shamrock" trinity - three bits growing out of an underlying stuff. In practice this is, roughly, how many unthinkingly view the trinity. Such a vulgar quaternity is rightly rejected by theologians. It can be seen immediately that the 'Godness of God' is considered at a completely different level to the three Persons in their roles and relations. What makes God God is fundamentally impersonal attributes that may be expressed in the Persons but not constituted by their mutual inter-play. So we can safely reject this version of things.
But I find that many theologians, having rejected the vulgar quaternity, congratulate themselves prematurely. There is also the insidious quaternity to be dealt with. There is another way of having a fourth...
Fundamentally this error consists in conceiving of the one God separately to a consideration of the three Persons in communion. Recently I read a theologian say "God is both one and three - both a person and a community." This is an example of the insidious quaternity. One-ness and Three-ness are laid side by side to uphold a belief in the equal ultimacy of one and three. Yet the one-ness of God is conceived of as a uni-personal one-ness - that is, it is separately considered to the multi-personal three-ness. One and Three were not mutually interpreting truths but instead the 'one God' is thought of in non-communal (that is, non trinitarian) terms.
This is the approach taken by by so many doctrine of God text books where De Deo Uno (on the One God) is addressed prior to De Deo Trino (on the Trinity). Yet, unless the two section are integrated at the deepest levels then there is grave danger of a fourth thing - i.e. "God plus Trinity" or "God apart from Trinity."
When this theological method is followed, often (not always but most times) section one unfolds such that the Three Person'd interplay takes no meaningful part in the discussions of the attributes. Yet, typically, these attributes are asserted to be the virtue by which God is God. On this view it is still possible to discuss the 'Godness of God' without reference to the perichoretic life of the Three. Here One-ness and Three-ness are considered to be non-competing perspectives on the same God. This effectively means that it is possible to speak in non-triune terms about the living God. 'God', then, is not the same thing as 'the Three Persons united in love'.
This is also a quaternity. Just a more insidious one.
And the only way I can see to avoid this fourth thing is to side with the Cappodocians: God's being consists without remainder in the Three Person'd perichoresis .
The one-ness of God is not a simple divine essence but the very unity of the Three. The being of God is not an underlying substance (contra the vulgar quaternity). But nor is it a separately conceived essence (contra the insidious quaternity). Rather God's being is the very communion by which the Three are One.
Trinity is not a perspective on the one God. Rather the only God there is is trinity. And the only way to conceive of Him is in triune terms. 'God' is 'Trinity'. Unless this strict identity is maintained a fourth enters in.
Thus we must never conceive of the one God in any other terms than trinitarian ones. (Re-write the text-books!). God's being is in His communion (to use Zizioulas's phrase). His One-ness is in His communion. And (let's not forget) His Three-ness is in His communion - the Three are only who they are in this eternal perichoresis. To put it another way: God is love.
This is a re-working of an older post on One-ness and Three-ness.