One can find a similar Sethian/Ophite tetrad in the Nag Hammadi text, Eugnostos (See The Nag Hammadi Library, 1990, pp. 227-28) : 1- Unbegotten Forefather; 2-Self-Begetter; 3- Man of Depth; 4-Sophia. (see also Ophite Gnosticism, Sethianism and the Nag Hammadi Library, Tuomas Rasimus,Vigiliae Christianae, Vol. 59, No. 3 (Aug., 2005), pp. 235-263).
Coincidently, there's another somewhat obscure connection of the Arba-il with Gnosticism - the word 'Barbelo', has been conjectured to be derived from that term. See Wace, p. 714): https://books.google.ca/books?id=WDA2AQAAMAAJ&pg=PA714&lpg=PA714&dq=wace+barbelo+714&source=bl&ots=VSOJvuXx-B&sig=9-y1VeU_T-rOBTeVVVsmWS57LWg&hl=en&sa=X&ved=0CCsQ6AEwAmoVChMIptzA7q_5xgIVDxWSCh0qEw1a#v=onepage&q=wace%20barbelo%20714&f=false
HPB gives a Babylonian example (see Rawlinson’s The Five Great Monarchies of the Ancient Eastern World, Or, The ..., Volume 2, p.15)
In this case, the tetrad concept can be found as far back as Sumerian sources, which takes us back to the beginning of Near East religion (see the Epic of Gilgamesh, N.K. Sandars, Penguin, 1960, p. 164). A Sumerian creation fragment from Nippur has (Heidel, The Babylonian Genesis, 1945, 72): When Anu, Enlil, Enki, and Ninhursagga Had created mankind,...
For another example from Christian theology, one can turn to Alchemy. Jung calls this concept the ''Three and the One'' Alchemical quaternary and he cites a text by a certain Orthelius in relation to the above image (from the Rosarium philosophorum 1550 , (Jung, p. 429)):
''There are said to be two treasures: one is the written word and the other is the word become fact (verbum factum). In the verbum scriptum Christ is still in swaddling clothes in his cradle (in cunis suis involutus); but in the verbum dictum et factum the word is incarnate in God’s creatures, and there, in a manner of speaking, we may touch it with our hands. From them we must raise up our treasure, for the word is nothing other than the fire, the life, and the spirit which the Holy Trinity did scatter abroad from the beginning of creation, and which brooded (incubavit) on the face of the waters, and which was breathed into (inspiratus) all things by the word of God, and embodied in them, as it is written: ‘’The spirit of God filled the whole world.’’ Some have expressed the opinion that this world spirit (spiritus mundi) was the third person of the Godhead; but they have not considered the word ‘Elohim,’’ which, being plural extends to all persons of the Trinity. They say this spirit proceeded from thence and was by it created, that I became corporeal and is the chief constituent of the Saviour (salvatoris) or Philosophical Stone, and is the true medium whereby body and soul are held united during our life.'' Jung, C.G. (1968). Psychology and Alchemy, Collected Works of C.G. Jung, Volume 12, Princeton, N.J.: Princeton University Press. pp.428-29
For the Kabalistic tetrad, in the Sefer Yetzirah 1, 9. I think de Zirkof adds an appropriate reference to 1, 12 which would refer to the tetrad of one, air, water, fire. She no doubt has her reasons using that reference for what she terms the abstract kabalistic trinity. Interestingly, Aryeh Kaplan gives a very similar, albeit more straightforward interpretation in his commentary of the Sefer Yetzirah (1997, 71) 1, 9 he refers to a kabalistic tetrad of Keter, Chakmah, Binah and Ruach Hakodesh. "This 'Holy Spirit' can be seen as the intermediate between Voice and Speech. It is thus also intermediate between Chakmah and Binah consciousness. Ruach HaKodesh is the divine inspiration and information that one can bring back from a state of Chakmah consciousness to one's normal state of Binah consciousness. Such Ruach HaKodesh is like Keter, which stands between Chakman and Binah, but which is above them. Both Chakhmah and Binah are functions of the mind itself, while Ruach HaKodesh is the 'breath of God' mentioned in the verse, 'I will fill him with the Breath of God, with Wisdom, Understanding, and Knowledge.'
To sum up, below is a chart of the concept of the mystic four that takes into account original sources that have come to light more recently. I think it remains a fairly obscure concept, but seems to have had some noticeable presence in ancient theologies, with a deep theosophical significance and so from Rawlinson to Inman to Blavatsky and Jung, it can be considered a bona fide theme in comparative religion, with Blavatsky maintaining that there is a spiritual symbolism common to them all:
1 - Brahma
2 - Siva
Man of Depth
4 – Devi Mahatmya