Corroborating Tolkien’s claim that the distinction within Eru is already to be discerned in the Ainulindalë is the interpretation Paul Kocher gave in 1985, eight years before the publication of the Athrabeth and its commentary in Morgoth’s Ring. As Kocher argued then, in calling the One “Ilúvatar,” or “Sky-father,”
Tolkien is suggesting that Ilúvatar, as Father, is merely the most active member of a more complex deity known as Eru, who is the only Lord of all… [T]he description of the Flame as “Imperishable” gives it divinity in its own right. Only God is eternal and cannot die. The Flame is not a possession of Ilúvatar but is co-equal with him in the Being of Eru the One… God the Father is the maker of the universe, but the Flame Imperishable, his coadjutor, infused it with life. Surely the “Secret Fire” or “Flame Imperishable” which “giveth Life and Reality” is very much like the Holy Spirit which works in the New Testament miracles underlying the whole Christian faith?
As it turns out, Tolkien himself answered Kocher’s rhetorical question in an interview he gave with Clyde Kilby, in which he divulged that the Secret Fire was none other than the Holy Spirit. Thus, while there is certainly no explicit awareness of or (again, consistent with St. Thomas) rational inference to the triunity of God in Tolkien’s fiction, what we do see, especially in the case of Finrod, is an instance of philosophical reasoning—ever so dimly illumined by the conceptual possibilities opened up by the “revelation” or rather “rumour” of those of the “Old Hope”—arriving at some “trace” of this divine mystery. As Tolkien himself describes the similar kind of deliberate yet significant ambiguity surrounding the identity of the wizards in The Lord of the Rings, “I have thought it best in this Tale to leave the question a ‘mystery’, not without pointers to the solution” (L 190).
[For another passage in which Tolkien connects the Holy Spirit and the Secret Fire, see here.]
 Kocher, “Ilúvatar and the Secret Fire,” 36-7.
 Kilby, Tolkien and the Silmarillion, 59.
 While I have focused on the image of the Flame Imperishable, Zimmer has seen a possible allusion to the Word of God, the second person of the Trinity, in Eru’s speaking the word Eä! that brings the world into being: “While in non-Christian-Neoplatonic thought the eternal act that constitutes the divine mind is termed the ‘nous,’ in the Christian tradition the nous is reformulated as the Word. Like the nous, the Word contains within itself all of creation as it exists eternally in the form of archetypal ideas… This double quality of the Word as both the intelligible structure and the willed act of creation is expressed in the Silmarillion by the single word ‘Eä,’ which means both ‘It is’ and ‘Let it be’…” Zimmer, “Creation and Re-creating Worlds with Words,” 53.