The Supersoul in Bhagavad-gita

Here are some qualities of this form of the Transcendental Lord, as given in Vedic literature:

The Supersoul

Is found in the heart of every being
Accompanies every being in the Game of Creation (lila shriti)
Is the Witness of the activities of every being (giva)
Is the Maintainer of all beings
Is the owner of all beings
Is the Supreme Controller
Is the One Who gives permission to the body of each being to act
It is the Giver of Remembrance, Knowledge and Forgetfulness
Does not taste the results of the activities of the bodies of beings

In the last chapter of the Bhagavad-gita (Bg. 18.61) Shri Krishna reminds that in the form of the Supersoul, He is in the heart of all.

īśvaraḥ sarva-bhūtānāḿ hṛd-deśe 'rjuna tiṣṭhati
bhrāmayan sarva-bhūtāni yantrārūḍhāni māyayā (Bhagavad-gita (Bg. 18.61)

O Arjuna, the Supreme Controller is- arjuna īśvaraḥ tiṣṭhati
in the heart area of all beings -hṛd-deśe sarva-bhūtānāḿ
and makes all beings travel-sarva-bhūtāni bhrāmayan
into a machine (gross and subtle material body) made of illusory energy - arūḍhāni yantra māyayā

The gross body made up of the five kinds of elements: earth, water, fire, air and space.
The subtle body made up of the three types of subtle elements: mind, intelligence and False Ego

Shri Krishna says to this effect in Bhagavad-gita 15.15:

sarvasya cāhaṁ hṛdi sanniviṣṭo mattaḥ smṛtir jñānam apohanaṁ ca
vedaiś ca sarvair aham eva vedyo vedānta-kṛd veda-vid eva cāham

I am in the heart of every being, - aham sanniviṣṭaḥ hṛdi sarvasya
from Me come remembrance, Knowledge and Forgetfulness. - as mattaḥ smṛtiḥ jñānam apohanam
The goal of the Vedas is that I may be known - ca vedaiḥ ca sarvaiḥ ham eva vedyaḥ
I am the real creator of Vedānta (the Conclusion of Knowledge) and the Knower of the Vedas - vedānta-kṛt veda-vit eva ca aham

Veda in Sanskrit means Knowledge, not Faith.

upadraṣṭānumantā ca bhartā bhoktā maheśvaraḥ
paramātmeti cāpy ukto dehe 'smin puruṣaḥ paraḥ (Bhagavad-gita 13.23)

In the material body there is a higher puruṣaḥ (one who enjoys pleasure) - dehe 'smin puruṣaḥ paraḥ.
He is also called the supersoul, parama-atma: - as apy ukto paramātma iti
The one who is the witness, who gives permission - upadraṣṭa anumantā
who maintains, is the owner and Supreme Controller. - bhartā bhoktā maha iśvaraḥ