The imperishable yoga, handed down from the sun through the first kings.

Advaita Vedānta

These commentators read 'this yoga' as the establishment in knowledge that is reached by way of selfless action, and they spell out a social purpose for teaching it to a king first. Krishna gave it to Vivasvat, the foremost of kshatriyas, to strengthen the warrior-rulers, so that, armed with the power of yoga, they could protect the priestly order; and when the priestly and warrior orders are both protected, the whole world is preserved. On 'avyaya', one of them describes the sun as the golden person seen within the solar disc named in scripture, and reads avyaya as 'of unbroken tradition,' hence beginningless. The imperishability of the fruit is also defended against the objection that any fruit to be achieved must be perishable, by appeal to the scriptural promise of non-return.

Śaṅkarācārya · Śrī Ānandagiri · Dhanapati Sūri · Śrī Nīlakaṇṭha

Advaita Vedānta

One commentator here explicitly rejects reading 'avyaya' as 'unbroken' or 'uninterrupted.' He argues that such a reading clashes with the very next words of the text, where it is said that with the long passage of time the yoga was lost. So 'avyaya' must mean imperishable in the sense that its fruit and its Vedic root do not perish, not that the chain of teaching was never broken.

Dhanapati Sūri

Viśiṣṭādvaita

These commentators insist the teaching Krishna recalls was never meant merely to spur Arjuna to fight. He declared it at the very beginning of an aeon to Vivasvat as the means to liberation, the highest human goal, for the deliverance of the whole world; through Manu, Ikshvaku, and the royal sages it came down by an unbroken succession of handing, until by the dullness of one hearer after another it became all but lost. One of them argues at length that 'this yoga' must be karma-yoga, since reading it as jnana-yoga misfits the surrounding context and contradicts the later command 'do action' at 4.15; he also notes that 'I taught it' implies there was no one else who could even know or speak it, and reads 'avyaya' as the Lord's eternal omniscience or, by way of fruit, the unperishing teaching.

Rāmānujācārya · Vedānta Deśika

Bhedabheda

This commentator keeps to a spare reading: Vivasvat is the sun, and Krishna, the imperishable one, declared the yoga to him. Its eternal character follows simply from its being enjoined by the eternal Veda. He then traces the chain to Manu, the wielder of the rod of justice, and to Ikshvaku, the first king.

Śrī Bhāskara

Dvaita

These commentators read the recital of the lineage as making a practical point that goes beyond mere history. The verse shows that 'this dharma was practised in earlier times' by the ancients; it is not just the succession of the teaching that is told, but their actual practice of it. The intent is to make clear that this same withdrawn dharma is to be practised by Arjuna too. One of them carefully distinguishes this from an earlier verse about a householder's conduct: there ordinary conduct was urged so that one's own proper action would not be abandoned, while here right conduct is spoken of with reference to the practice of the withdrawn dharma.

Madhvācārya · Śrī Jayatīrtha

Śuddhādvaita

For these commentators the lineage is the path's first credential of grace. The speaker of the yoga is Vishnu himself, from whom the sun arose; the path is received not as a private discovery but as the Lord's own gift, handed down through the sun-king Vivasvat, through Manu to the Ikshvakus, in whose very line Ramachandra himself stood. The credential is grace, not novelty. One of them adds that the yoga is indestructible and fruit-bearing precisely because it generates relation with the Lord, and that the lineage is set at the head of the chapter so the disciple may know the teaching is no contrivance of the hour but a stream of grace handed down from the Lord himself.

Vallabhācārya · Śrī Puruṣottama

Bhakti

These commentators frame the chapter as Krishna praising the very yoga of knowledge, accomplished through desireless action, that the prior chapters taught. One hears in the recitation of the chain the assurance that what follows is no novelty of the moment but the ancient royal tradition itself, set going from the Lord and never stepping outside the line of dharma-knowing kings. They note that Krishna is about to expand the teaching by laying down the offering of works to Brahman and the discrimination of the great Vedantic terms, and that here he first establishes its standing as 'lineage-received.' One reads avyaya as eternal because, being the meaning of the Veda, it does not perish, and because its own fruit never fails.

Śrīdhara Svāmī · Śrīla Viśvanātha · Śrīla Baladeva

Modern

These commentators read 'this imperishable yoga' specifically as karma-yoga. One reasons that the sun, Manu, and Ikshvaku were all householders who reached the supreme attainment while still living the householder life and seated on the throne, so the royal-lineage context fits karma-yoga most naturally; karma-yoga is therefore the most ancient discipline, eternal in its truth and royal in its line, not a fallback for those who cannot manage knowledge or devotion. He takes 'avyaya' to establish the very eternality of karma-yoga, since the jiva's relation with the Lord is itself eternal, and he draws a contemplative aim from 'Vivasvat': as the sun is always working yet, lighting everything, remains untouched, so the seeker should do his allotted duty and remain unstained. The other reads avyaya as inexhaustible, permanent, untouched by past, present, or future, and likewise glosses the yoga taught to the sun as karma-yoga.

Swami Ramsukhdas · Lokmanya Tilak

Modern

This commentator gives the plain historical and social sense without sectarian elaboration: Vivasvan is the sun, Ikshvaku the son of Manu and reputed ancestor of the solar dynasty of kshatriyas. The yoga is called imperishable because the liberation attainable through it is imperishable. And Krishna taught it to the sun-god at the beginning of evolution so that rulers possessing this knowledge could protect the priestly class and govern their kingdoms with justice.

Swami Sivananda

Kashmir Shaivism

This commentator's remarks fall mainly on the verses just after, but bear on the lineage here. By noting that this knowledge, though it had come down through the succession of teachers, is in our own day lost, the Lord shows the rarity and the weightiness of this knowledge; its very disappearance marks how precious and grave it is.

Ācārya Abhinavagupta