In SD I, 280 we find that by HPB the “Causeless Cause of All Causes” is identified with kāraṇa:

The ever unknowable and incognizable Karana alone, the Causeless Cause of all causes, should have its shrine and altar on the holy and ever untrodden ground of our heart — invisible, intangible, unmentioned, save through “the still small voice” of our spiritual consciousness.

As we have seen in The footnote in SD I, 14-15, the “Causeless One Cause”, the “Rootless Root” is the unmanifested Logos, which we have called the First Logos. (see The Three Logoi)

In SD I, 41 (explaining stanza I śloka 5) is stated that in the period of pralaya, when the universe has returned to its “one primal and eternal cause”, that

“Karana” — eternal cause — was alone.

In SD I, 93 we find in stanza IV śloka 4 the “eternal nidana”, or nidāna, which is a Sanskrit word for cause, the first cause in particular, or the cause of existence (cf. Monier-Williams), which in stanza IV śloka 5 is identified with “’DARKNESS,’ […], ADI-NIDANA SVABHAVAT”. In the note in SD I, 93n we find an explanation of the word nidāna:

* […] but in this instance, it is a term to denote the ceaseless and eternal Cosmic Motion; or rather the Force that moves it, which Force is tacitly accepted as the Deity but never named. It is the eternal Karana, the ever-acting Cause.

Here, nidāna is identified with kāraṇa, and with the “force” resulting in cosmic motion. The concept of abstract motion is, together with abstract space and abstract duration, one of the central concepts in the esoteric philosophy presented in The Secret Doctrine. In the Book of Dzyan, this unmanifested aspect behind cosmic motion is symbolised as the great breath, while cosmic motion itself is called the divine breath.

In SD II, 46 we find out some more about kāraṇa, in a quotation from the “Commentary”:

“After the changeless (avikâra) immutable nature (Essence, sadaikarûpa) had awakened and changed (differentiated) into (a state of) causality (avayakta), and from cause (Karana) had become its own discrete effect (vyakta), from invisible it became visible. The smallest of the small (the most atomic of atoms, or aniyâmsam aniyâsam) became one and the many (ekanekárûpa); and producing the Universe produced also the Fourth Loka (our Earth) in the garland of the seven lotuses. The Achyuta then became the Chyuta.*

We see that kāraṇa itself changes into its own effect, which is called vyakta, a term generally indicating that which is manifested, or the manifested universe, but another one of its meanings (as an adjective) is visible, apparent or caused to appear.  (Monier-Williams)

In the Viṣṇupurāṇa (VP), in the 1840 translation of Horace H. Wilson, which was regularly consulted by HPB, we find in Book I chapter II page 8, in Wilson’s notes, explanations of the Sanskrit terms from the quotation of the Commentary:

2. This address to Vishńu pursues the notion that he, as the supreme being, is one, whilst he is all: he is Avikára, not subject to change; Sadaikarúpa, one invariable nature: he is the liberator (tára), or he who bears mortals across the ocean of existence: he is both single and manifold (ekánekarúpa): and he is the indiscrete (avyakta) cause of the world, as well as the discrete (vyakta) effect; or the invisible cause, and visible creation.

[…]

4. Ańíyánsam ańíyasám, ‘the most atomic of the atomic;’ alluding to the atomic theory of the Nyáya or logical school.

5. Or Achyuta; a common name of Vishńu, from a, privative, and chyuta, fallen; according to our comment, ‘he who does not perish with created things.’ The Mahábhárata interprets it in one place to mean, ‘he who is not distinct from final emancipation;’ and in another to signify, ‘exempt from decay’. A commentator on the Káśikhańd́a of the Skánda Puráńa explains it, ‘he who never declines (or varies) from his own proper nature.’

What it means that we find these terms here in one page in Wilson’s notes is, I think, open for debate.

In the text of the Viṣṇupurāṇa (VP I.II.1-5) we can try to identify the terms from the quotation of SD II, 46:

The idea of the Causeless Cause, or the cause, kāraṇa, becoming its own effect, vyakta, is formulated by Wilson in note 3 on page 8:

The world is therefore not regarded by the Pauranics as an emanation or an illusion, but as consubstantial with its first cause.

Of course much more could be said about this passage in the VP, relating to the stanzas from the Book of Dzyan, an example being that in VP I.II.4, Viṣṇu is called mūlabhūta, the root of the world (Wilson), a term found in stanza II śloka 1 (SD I, 53).

Returning to our theme here, we might turn to another location in the stanzas, in SD I, 107-108, stanza V śloka 2:

2. […] (a). THE DZYU BECOMES FOHAT; […] RUNS CIRCULAR ERRANDS. […] TAKES THREE, AND FIVE, AND SEVEN STRIDES THROUGH THE SEVEN REGIONS ABOVE AND THE SEVEN BELOW (the world to be). HE LIFTS HIS VOICE, AND CALLS THE INNUMERABLE SPARKS (atoms) AND JOINS THEM TOGETHER (c).

In HPB’s extensive commentary to (c) we find (in SD I, 109):

When the “Divine Son” breaks forth, then Fohat becomes the propelling force, the active Power which causes the ONE to become TWO and THREE — on the Cosmic plane of manifestation. The triple One differentiates into the many, and then Fohat is transformed into that force which brings together the elemental atoms and makes them aggregate and combine.

and (in SD I, 110):

By the action of the manifested Wisdom, or Mahat, represented by these innumerable centres of spiritual Energy in the Kosmos, the reflection of the Universal Mind, which is Cosmic Ideation and the intellectual Force accompanying such ideation, becomes objectively the Fohat of the Buddhist esoteric philosopher. Fohat, running along the seven principles of AKASA, acts upon manifested substance or the One Element, as declared above, and by differentiating it into various centres of Energy, sets in motion the law of Cosmic Evolution, which, in obedience to the Ideation of the Universal Mind, brings into existence all the various states of being in the manifested Solar System.

Combining the phrase “THE DZYU BECOMES FOHAT “ from stanza V śloka 2 with this last quote, we must conclude that the dzyu is identical to “the reflection of the Universal Mind, which is Cosmic Ideation and the intellectual Force accompanying such ideation”. Dzyu becomes fohat “when the ‘Divine Son’ breaks forth”, i.e. at the moment the universe comes into manifestation, so we can conclude that dzyu is the unmanifested principle which is at the basis of fohat, the (manifested) “propelling force” which “sets in motion the law of Cosmic Evolution”. This principle is of course kāraṇa, which is, as we have seen, the “force” resulting in cosmic motion, or the principle of abstract motion, in the Book of Dzyan symbolised as the great breath.