The term so'ham occurs in many upanishads like Isha upanishad etc.
How do major schools like Advaita, Dvaita, Vishistadvaita and Achintya Bhedabheda interpret that?
In the ĪśaU, the passage occurs in verse 16:
pūṣann ekarṣe yama sūrya prājāpatya vyūha raśmīn samūha tejaḥ; yat te rūpaṁ kalyāṇatamaṁ tat te paśyāmi yo ‘sāv asau puruṣaḥ so ’ham asmi.
The interpretation of Sankara, VedantaDesika and Madhva mainly concern the ontological relationship between what is denoted by the term saḥ and what is denoted by the term aham in the passage
According to Śaṅkara, the last four verses of the ĪśaU comprise the prayer of a person to the deity in the sun at the time of his death and he thereby becomes the deity himself. Śaṅkara adds that the deity to whom the prayer is offered is lower Brahman (aparabrahman). The first pāda of verse 16 is a series of vocatives addressing the deity in the sun. Śaṅkara identifies the deity as the nourisher, the lone traveller, the controller, the sun, and the son of Prajāpati.Then he gives the following explanation on the fourth pāda, yo ’sāv asau puruṣaḥ so ’ham asmi, which includes the phrase so ’ham:
yaḥ asau, the Person who is there, in the solar orb [. . .], and who is called a Person (Puruṣa) because he has the figure of a person: [. . .] saḥ aham asmi, that Person am I.
Śaṅkara gives ‘I become (bhavāmi)’ as a gloss of asmi. It is clear that, according to him, the passage suggests the oneness of the person offering the prayer with lower Brahman, who is the object of the prayer
In contrast to Śaṅkara who says that the person who is the object of the prayer is lower Brahman, Vedāntadeśika says that the person is the supreme self (paramātman). Commenting on the fourth pāda of verse 16, Vedāntadeśika first says that the repetition of asau expresses great respect. Śaṅkara did not offer any interpretation on this point. Then Vedāntadeśika explains that puruṣa in the fourth pāda indicates the supreme person (i.e. Vishnu), ‘Puruṣa (indicates) the great person, who has qualities such as completeness and being the old existence, who has a special body with the colour of the sun, who is famous in the Puruṣasūkta and so on, which are dedicated to none other than He who is mentioned in all the Vedas (i.e. Vishnu)’
Then Vedāntadeśika explains that the word aham in the pāda indicates the supreme self within (paramātman) and does not refer to the individual self (jīva),
[The verse says] so ’ham asmi. Through the individual self [jīva], the word ‘I [aham]’ here indicates the inner self [i.e. paramātman]. Precisely because of this, the first person copula ‘am [asmi]’ also refers to the supreme self [paramātman], who is characterized by the individual self, [the individual self] who has a separate nature. For, exactly this is taught [by Pāṇini as] asmady uttamaḥ. When the word ‘I’ indicates the Supreme Self through the individual self, still [the use of] the first person copula does not cease
According to Vedāntadeśika, ‘I’ (aham) indicates the Supreme Self who is distinct from the individual self. If that is the case, it may be asked why the first person copula ‘am’ (asmi) is used. In other words, the third person copula ‘is’ (asti) may be more appropriate if ‘I’ actually denotes the third person, the Supreme Self. To answer the question, Vedāntadeśika refers to the Aṣṭādhyāyī 1.4.107: asmady uttamaḥ. This rule teaches that when the first person pronoun asmad, which is the root form of aham, is used, the verb must have the first person ending (uttama). In conclusion, according to Vedāntadeśika, the fourth pāda ‘yo ’sāv asau puruṣaḥ so ’ham asmi’ means that venerable puruṣa (Vishnu), who is mentioned in the Puruṣasūkta and so on, is identical with the supreme self (paramātman) within.
He comments on the first and fourth pādas of verse 16 (pūṣann ekarṣe yama sūrya prājāpatya […] yo ’sāv asau puruṣaḥ so ’ham asmi) as follows:
Since he has the most excellent knowledge, Vishnu is known as Ekarṣi. Hari is (known as) Yama because he is the controller. (He is known as) Sūrya because he is accessible to the learned people. (He is known as) Prājāpatya because he is especially accessible to Prajāpati. And He (is known as) Aham because he cannot be rejected.10 (He is known as) Asmi, abiding in all the jīvas, because he is proud of his eternal existence. But supreme Hari Himself is separate from all the jīvas.
Here, Madhva is explaining the verse in the way of nirukta, that is, a word packed with semantic meanings based on phonetic similarity. Vishnu is called Yama because he controls (niyamanāt). He is called Sūrya because he is accessible to the learned people (sūrigamyatvāt). He applies the same method in interpreting the phrase so ’ham asmi. Instead of explaining aham as the first person pronoun and asmi as a copula, Madhva says that they are both secret names of Vishnu. According to him, Vishnu is called Aham because he cannot be rejected (aheyatas) and Asmi because he is proud of his eternal existence (nityāstitamānāt) According to Madhva, therefore, the words aham and asmi in the phrase so ’ham asmi have nothing to do with the individual self. Furthermore, explaining the repetition of asau which occurs in yo 'sāv asau puruṣaḥ of the fourth pāda, he says that one of them is a locative of asu, which means the life air (prāṇa). The entire fourth pāda would then mean that the person in the life air is Vishnu, who cannot be rejected and who is conscious of his own eternal existence.
Though Baladeva Vidyabhusana commented on Isopanishad, he didn't explain this phrase. However, there occurs a similar phrase gopālo ’ham ('I am Gopala') in Gopala Tapani Upanishad which he explained in his commentary to Brahma Sutras(3.3.46)
While pre-decessors offered ontological interpretations of the passage, Baladeva takes an entirely new approach based on the Gauḍīya aesthetic theory of rasa.
Baladeva explains that the meditation so ’ham is a type of devotion because of the context of the GoTāU in which the phrase appears. In his GoBhā, Baladeva examines this context of the Upanishad by applying the method of the sixfold criteria (tātparyaliṅga), which is an exegetical tool employed by Vedantists to determine the meaning of texts.
Alluding to the state of the gopīs described in the tenth book of the BhāPu, Baladeva explains that extreme attachment to Krishna can create a mental state where a devotee identifies herself with him, thinking ‘I am Krishna’
The expression ‘I am Krishna (krṣṇo ’ham)’ occurs in Bhāgavata Purana 10.30.19: ‘Another [gopī], with [her] mind intent on Him [i.e. on Krishna], placing her arms on another [gopī] and walking, said, “I am Krishna. Behold [my] lovely gait!”’
According to the Gauḍīya tradition, this emotional absorption in Krishna as a result of one’s separation from him is called vipralambha rasa. This sentiment is considered the highest expression of devotion to god.
Baladeva selects different passages from the GoTāU which fulfil the sixfold criteria:
(1) The opening and concluding sentences GoTāU 1.14, 2.63 (2) Repetition GoTāU 2.17 (3) Unprecedentedness (4) reasoning GoTāU 1.2, 1.3, 1.6, 1.7, 1.11, 1.14 (5) Fruit GoTāU 1.5, 1.23 (6) Praise GoTāU 1.33
After citing the passages from the GoTāU and explaining how they fulfil the sixfold criteria, Baladeva concludes that the phrase so ’ham / gopalo'ham, which occurs in the GoTāU, teaches a type of devotion.