Verse 7

7.  That is known as the fourth quarter: neither inward-turned nor outward-turned consciousness,  nor the two together; not an undifferentiated mass of consciousness; neither knowing,  nor unknowing; invisible,  ineffable,  intangible,  devoid of characteristics, inconceivable,  indefinable,  its sole essence being the consciousness of its own Self; the coming to rest of all relative existence; utterly quiet; peaceful; blissful; non dual; this is the Atman,  the Self; this is to be realised.

This state of consciousness is closest,  within Creation,  to the infinity of the Absolute.
This consciousness is the  "Absolute Self"  of the three states of  "self"  waking,  dreaming and sleeping.
This  "Self"  in the so called  "fourth"  is the silence and stillness that underlies and follows the dissolving of all other states of material existence  (AUM)  back to unmanifest Knowledge,  either of the forms of self or of the forms of the Universe or Om.

"That is known as the fourth quarter"

As explained at the very beginning of this Upanishad the  "quarters"  do not exist in reality.  The reality of Existence is that unchanging immutable Absolute Consciousness alone Exists.  This immutable existence suffers no division through quarters or any other form of differentiation.   The describing of quarters is to describe the experiencing in consciousness of the knowledge of that unchanging Absolute.
As the ocean is always the one ocean of water so the Absolute is always the ocean of absolute knowledge as Consciousness.
Sometimes the Ocean is witnessed as being blue or green or grey but is in reality unchanging.  Just so,  the witnessing experience of consciousness is sometimes of waking or dreaming or the experience of the non-experience of deep sleep but the reality is that these states are always and only an experience.   The Absolute is unchanging in all ways.
So this  "fourth quarter"  describes The stillness of the Absolute Self.  With or without the experiencing of  "quarters"  or states the Absolute Self remains unchanging.  Even when there is the experience of waking sleep,  dream sleep or deep sleep that experience is merely the reflexion of the light of knowledge on Consciousness as is the blue green or grey a reflexion of light on the ever present and unchanging same ocean of water.

The true colour of the ocean will be seen only through absolute white light.
The true singularity of consciousness will be seen only through Absolute Truth.

"neither inward-turned nor outward-turned consciousness,  nor the two together"

Because we are talking of  "Self"  Consciousness,  we are talking of material existence without a material awareness,  "neither knowing,  nor unknowing"  The only consciousness there is,  is consciousness unaware of its own transient state  (being the  "Self"  of creation). This is the reality of Non-Duality.
Consider   :-
Brihadaranyaka Upanishad 4.  5.  15  Because when there is duality,  as it were,  then one sees something. 

"not an undifferentiated mass of consciousness; neither knowing,  nor unknowing;"

Because the Absolute is literally Absolutely all Knowledge as Consciousness which is the totality of Existence itself it transcends name and consideration as a  "mass",  it just Is.
The Absolute does not   "know"  because being all knowledge there is nothing to know.
The Absolute is not   "unknowing"  because being all knowledge there is not that which is unknown.
The Absolute is Knowledge itself alone,  being Non-dual there is not that other to be known or unknown.

"invisible,  ineffable,  intangible,  devoid of characteristics,  inconceivable,  indefinable"

The infinitely subtle Absolute can never be visible nor in the Non-Dual reality of existence is there the duality of seer and seen,  Nor the defined or the definable.  Being unknowable the Absolute is inconceivable because the truth of the Absolute cannot be imagined by anything less than that Absolute Reality.

"its sole essence being the consciousness of its own Self"

This statement has the meaning that  "its own Self"  is Consciousness alone,  Consciousness is its  "essence".  The Absolute cannot be not conscious of its own existence because it is this very non-dual existence itself.  Likewise the Absolute will not be particularly conscious of its own existence  "Self"  because there is no such known as non-existence.

"the coming to rest of all relative existence; utterly quiet; peaceful; blissful; non dual".

The Absolute in Truth is the Singularity.  There is nothing eternally relative in Non-Duality.  This statement describes the state of Existence at rest in unmanifest Knowledge after the return of the transient relative universe,  therefore now resting in its eternal natural state of Truth Consciousness and Bliss.

"this is the Atman,  the Self; this is to be realised".

The Absolute is the Truth and support of the concepts  "Atman"  or  "Self".  The statement  "this is to be realised"  declares the highest intention and purpose of all the scriptures.    It is said by the Sages that to know the one thing is to know all things.  This Realisation of the Absolute is that one thing that is the highest truth."This is to be realised".

This Reality of the Absolute Self is being described here because the three imagined quarters or states of consciousness relative to the created have now been described,  they are waking sleep,  dream sleep  and dreamless sleep,  or Vaisvanara,  Taijasa  and Prajna.
The said fourth state is the experience,  as it were,  of the Unity of the Absolute.  This unity occurs only through Self Realisation or after the dissolution of the said three "quarters" of AUM or the Universe back to unmanifest knowledge which is described next through verses 8-12.