The shadow personifies everything that the subject refuses to acknowledge about himself and yet is always thrusting itself upon him directly or indirectly— for instance, inferior traits of character and other incompatible tendencies. - Carl Jung, The Archetypes and the Collective Unconscious (1959)
Carl Jung defines the Shadow as those parts of the psyche and identity that the individual rejects. Now let us apply this theory to Lorkhan within the frame of his relation to Akatosh.
the madness of the Time God and the first challenge of his shadow, who in nothingness saw those endless possibilities first - The Nine Coruscations
In the aetheric thunder of self-applause that followed (nay, rippled until convention, that is, amnesia), is it any wonder that the Time God would hate the same-twin on the other end of the aurbrilical cord, the Space God? That any Creation would become so utterly dangerous because of that singular fear of a singular word’s addition: “I AM NOT”? - Et'Ada, Eight Aedra, Eat the Dreamer
Still others, like Fifd of New Teed, say that beneath the Pelinal's star-armor was a chest that gaped open to show no heart, only a red rage shaped diamond-fashion, singing like a mindless dragon, and that this was proof that he was a myth-echo, and that where he trod were shapes of the first urging. - Song of Pelinal
Akatosh and Lorkhan are often conflated with one another to the extent they are sometimes considered the same being. In fact, this is one of the preeminent themes in the Songs of Pelinal, however, the full exegesis of that work to prove this assertion goes beyond this essay..
In Kirkbride's Et'Ada, Eight Aedra, Eat the Dreamers - Akatosh's loathing for his Shadow comes from Lorkhan's addition of the word NOT to Akatosh's own self-affirmation of I AM. This 'I AM NOT' is a denial of the Shadow-self's own existence, and thus conflicts with the Actual-self's assertion of its own existence.
But why does the Shadow deny itself? We find the answer in the quoted passage of The Nine Coruscations; the Shadow, in the denial of its own existence, within the state of nothingness, found endless possibilities.
But what inspired the Shadow to even look at idea of self-denial?
Anu encompassed and encompasses all things. So that he might know himself he created Anuiel, his soul and the soul of all things. Anuiel, as all souls, was given to self-reflection, and for this, he needed to differentiate between his forms, attributes, and intellects. Thus was born Sithis, who was the sum of all the limitations Anuiel would use to ponder himself. Anuiel, who was the soul of all things, therefore became many things, and this interplay was and is the Aurbis. - Altmeri Heart of the World
When Anu broke itself, it did so to understand its nature. In its sundering, the values that swam in its vastness thought to know themselves. The et'Ada Gears gave themselves many names and set their will to building. - Truth in Sequence
There was first only Atak, the Great Root. It knew of nothing but itself, so it decided to be everything. It grew and grew, trying to fill the nothing with itself. As it grew it formed new roots, and those roots took names, and they wanted space of their own to grow. - Children of the Root
As you can see, multiple creation myths place the impetus of the Aurbis as being the result of a cosmic exercise of self-reflection and understanding. This is the 'godhead' that is often brought up in more esoterical lore discussions. This is the 'Cosmic-Self' that encompasses and begets the Aurbis. The et'Ada (the Aedra, the Daedra, Magna-Ge) are all attributes and aspects of the Cosmic-Self that have gained their own awareness.
'Look at the majesty sideways and all you see is the Tower, which our ancestors made idols from. / The secret Tower within the Tower is the shape of the only name of God, I.' - Vivec, Sermon 21
This isn't Vivec being egotistical, this is Vivec explaining that the Tower (I) is a symbol for the sense of self. The Majesty sideways is the Wheel and the Wheel is the Aurbis. Picture a wheel in your mind and turn it towards yourself so that the spokes are no longer visible and only the rim is. As the Wheel spins faster and moves farther, the only shape you can make out is a vague vertical form.
This is the Tower, this is 'I'.
Thus, the Aurbis is a Secret Tower in the shape of the letter 'I', which is the pronoun used to express the self. Ergo, the Aurbis is an expression of an identity/ego: the Cosmic-Self.
The Aedroth Aka, who goes by so many names as to perhaps already suggest what I’m about to commit to memospore, is completely insane. His mind broke when his “perch from Eternity allowed the day” and we of all the Aurbis live on through its fragments, ensnared in the temporal writings and erasures of the acausal whim that he begat by saying “I AM”. -  Et'Ada, Eight Aedra, Eat the Dreamer
The et'Ada Gears named each and each, in their way. Our lessers see this as a kindness, but the Mainspring Ever-Wound calls it a curse, rooted in selfish pride. To name is to cleave one from another. It is the death of Anuic convergence and the Nirn-Ensuing—the misassembled Dragon that breathes dry falsehood and whose name is "Multitude." - Truth in Sequence, Vol 2
"I am Kena Warfel Tomasin, and I can prove that Akatosh, Nirn, and Oblivion are one," said Warfel, writing out the mathematical formula that showed it was so. - The Four Suitors of Benitah
Linear time layered atop infinite possibility, thus did Aka … in the South, and yet … learned why his insanity is all that is and could be. … by this lesson … Ada-mantia, stable spire fixed by a stone of nothing-possible - The Nine Coruscation
The spike of Ada-Mantia, and its Zero Stone, dictated the structure of reality in its Aurbic vicinity, defining for the Earth Bones their story or nature within the unfolding of the Dragon's (timebound) Tale. - Aurbic Enigma 4
Lorkhan saw the Wheel on its side and beheld the secret that the Aurbis was a Tower.
As shown in the text Aurbic Enigma, Towers tell narratives and, in this case of the Aurbic Secret Tower, that narrative is autobiographical.
In Aldmeri theology, Anu's soul is Anui-el. And Anui-el's soul is Auri-el. But is there a difference between the soul and the self? In the Aldmeri ruins of Torinaan, Anu is considered one of the Eight Divines. In Coils of the Father, it is Auriel the Dragon referred to as the Soul of Anu, not Anui-el. And Girnalin claims it was Anui-el that established the time laws in Nirn, not Auri-el. They are conflated as the same being, but expressed in metaphysical subgradients.
Anu/Anui-el/Auri-el/Akatosh/Alkosh/Satak/Satakal/AKHAT/Akha, call it whatever you will; it is the same being, the same Cosmic-Self. But they are different expressions of the same Cosmic-Self in accordance with different archtypes within the narrative: we call this mythopoeia, the making and shaping of myths.
Therefore, I would assert that the Aurbis is the Time-Dragon's narrative of itself and all the spirits of the Aurbis are just fragments of the Time-Dragon: a living, ever-changing autobiography.
When Lorkhan saw the Wheel on its side and beheld the Secret Tower, he understood this reality: there is only the Dragon, and all else emanates from him.
In Tasawwuf (Sufism) teachings of Islam, various Sufi Shiekhs refer to this phenomena as 'conditional existence'. In his book, The Niche of Lights, Imam Al-Ghazali puts forth an allegory of a man who is surrounded by great luxury and wealth and he lacks nothing in life. However, all of his wealth is in reality borrowed from someone else and he owns nothing himself. Thus, can such a man actually be considered rich? In the same vein, where one's existence and identity are defined and reliant on another, can one be said to exist at all? As such, Sunni Islamic teachings express that existence is contingent upon Divine Will.
Imposing this notion on Lorkhan: we can infer that this is why Lorkhan added on NOT to I AM, as he understood that his existence is conditional based off of Akatosh's.
But now we must ask the question: what are endless possibilities Lorkhan saw in nothingness? But to do this, we must first explore the phenomena of 'nothing' within the Aurbis.
Late is the lover that comes to this by any other walking way than the fifth, which is the number of the limit of this world. The lover is the highest country and a series of beliefs. He is the sacred city bereft of a double. The uncultivated land of monsters is the rule. This is clearly attested by ANU and his double, which love knows never really happened. - Vivec, Sermon 35
Our lessers know the Source as two forms: Anu and Padomay, but this binary is without merit. One of the Lorkhan's Great Lies, meant to sunder us from the truth of Anuic unity. Our father, Sotha Sil, would have us know the truth: there is no Padomay. Padomay is the absence of value. The lack. A ghost that vanishes at first light. A Nothing. There is only Anu, sundered and known by many names, possessing many faces. The one. - Truth in Sequence, Volume 1
The first ones were brothers: Anu and Padomay. They came into the Void, and Time began. - The Anuad
The Fifth Walking Way is the Enantiomorph. The particulars of this Walking Way go beyond the scope of this essay, but Vivec associates the Enatiomorph with the number 2 in the Scripture of Numbers. These two are the Rebel and the King, whose conflict is solidified by the Witness Maimed. However, within the theme of this essay, let us posit that the Rebel and King are the Shadow-Self and the Cosmic-Self. Since they are both the same person, do they count as two separate people? Can an individual in conflict with oneself be considered two?
This is why Vivec says the sacred city is bereft of a double. And that Anu and his double never really happened. It is also why in Truth in Sequence Padomay is stated to not exist.
Let us consider: Anu is defined as everything. Padomay as defined as nothing. However, by definition, nothing cannot exist. Because nothing is the absence of existence. If Padomay does exist, then it it not nothing, it is something. And something is part of everything which is Anu. Thus, if Padomay exists, it is actually a fragment of Anu.
This is something that I have personally come to dub as the 'Padomaic Paradox'.
In the Anuad, it is interesting to note that Anu and Padomay come into the Void together. That is to say, Padomay is not conflated with the Void but as something that emerged seperately into it. But as soon as the two do come into existence, so too does Time. Time, of course, is Akatosh/Auri-el and too many other names to list.
Therefore, the impetus of Time is Anu and his double that was never really there. Within older iterations of the Monomyth, and in the current French translation, Satakal (who is compiled of the Yokudan versions of Anu and Padomay) is conflated with Akatosh. And in Children of the Root, the compilation of Atak and Kota (Anu and Padomay) is Atakota (Akatosh).
Thus, within certain interpretations, Akatosh is 
both Anu and Padomay.
"To be the Worldskin is to be everything, and to be everything is to be nothing." - Knowing Satakal
In Yokudan folk-tales, which are among the most vivid in the world, Satak is only referred to a handful of times, as "the Hum"; he is a force so prevalent as to be not really there at all. - The Monomyth
"Before creation, there was void. Nothing. But even nothing must change. Sithis sundered the nothing, creating the possibilities of something. These ideas bloomed and faded, as all things should. As all things shall." - Nisswo Uaxal
If Anu and Padomay are the same, if Everything and Nothing are the same; then in nothingness there is the potential to be everything.
If Akatosh as the Cosmic-Self is everything, and Lorkhan as his Shadow-Self is nothing, then Lorkhan too may be everything. But first, he must deny he exists as Akatosh, so that he may exist as something else.
Wait. Why would anyone want to purposely fail the process of CHIM?
And this is the most-reached destination of all that embark upon this road. Why would Lorkhan and his (unwitting?) agents sabotage their experiments with the Tower? Why would he crumble that which he esteems? Perhaps he failed so you might know how not to. - Vehk, Vehk's Teachings
"We curse you, noisy Lorkhaj, to walk Nirni for many phases." - Words of Clan Mother Ahnissi
"You are blood-made-glorious, uncle, and will come again, as fox animal or light." - The Adabal-a
Or perhaps he failed so that he could walk the Wheel as something other than Akatosh's Shadow, and then go beyond the Tower as something new?
TL;DR: Lorkhan is Akatosh's shadow psyche that's trying to break free and be its own existence... 
maybe.