sadbhāve sādhubhāve ca sadityetat prayujyate | praśaste karmaṇi tathā sacchabdaḥ pārtha yujyate ||17.26|| The word “sat” is used with reference to bringing (something) into existence and (with reference to) a righteous life; so too, Pārtha (Arjuna)! The word “sat” is used for a sanctifying karma.
The word sat is used in different senses, all of which are good. sadbhāve means bringing into existence something that was not there before. Śankara gives an example of the birth of a son. It is also used for any object that exists like a pot exists - ghatah san, the cloth exists - patah san, etc. Since it is used in the sense of existence, and the only existence is Parameśvara, the word sat is used to sanctify karma in religious context.
sādhubhāve means righteous life, a life lived in keeping with dharma. One who follows dharma or knows the sadvastu brahman is called satpurusah and the association with such person is called satsanga.
The word sat is used to denote brahman as the cause of creation, “sat eva somya idam agre āsīt ekam eva advitīyam - this world before creation was sat - existence alone, without followed by second.” (Chāndogyopanisad 8th chapter 6.2.1). This world which is name and form, is available for perception and transaction, was sat-existence alone which is not available for perception. When pure existence doesn’t associate with any object, there is no attribute which we can perceive and identify with. If it is not available for perception, how do we know it exists? Because that pure existence is me - ātmā alone, I cannot but know. Even in the deep sleep when there is no identification with the mind, I am pretty much there to recollect my existence in the very next morning. Therefore sat - existence alone is cit - consciousness which is self-evident.
The identification between ātmā and anātma is called superimposition - adhyāsa. But how ignorance which is the cause for superimposition can co-exist with self-effulgence ātmā which is knowledge? It is just like rope snake wouldn’t be born on the rope when bright light is there.
To answer that, we need to know that cit / caitanyam ātmā is having two aspects, samanya-caitanyam - general consciousness and viśesa-caitanyam - specific consciousness. General consciousness is the pure existence - sat which is the basis for manifestation of all names and forms. Since names and forms are not there, there is no identification with them, and it is not opposed to ajñāna - ignorance. This is why we experience ignorance and happiness in the deep sleep. Even though the mind and sense-organs are suspended in deep sleep, one who woke up from deep sleep recollects “I sleep happily, I don’t know anything.” In this manner the self-effulgent self recollected ignorance experience in itself. Therefore the nature of ātmā is not opposed to ajñāna, since it is the basis for all, lending existence and consciousness to all anātmā including ignorance.
Whereas specific consciousness manifests only through antah-karana-vrtti - thought of the mind which is opposed to ajñāna. Therefore the same and only caitanya-ātmā identifying with the individual mind becomes having individuality called jīva, is veiled by ignorance. Just like general fire in the form of fire principle which presents and pervades the fuel stick is not opposed to darkness because it is in an un-manifested state. That same fire alone, by rubbing the fuel stick on another wood, manifests into visible form, and lit it on the wick of the lamp. Being in specific nature, it becomes opposed to darkness. Since vrtti - thought is the main cause for consciousness me to be opposed to ignorance, therefore only by direct appreciation of vrtti “aham brahma asmi - I am brahman”, which is born through maha-vakya in vedānta teaching, then total destruction of ajñāna is possible.
When Om tat sat is uttered at the end of studying vedānta, we are ascertaining Om which represents the entire manifest and unmanifest world, and tat - that brahman who is the basis for the entire creation is sat - existence me who never ceases. Hence oneness between jagad-world, jīva-individual and īśvara is understood.