4.1 A lineage, named Krishna speaks

इमं विवस्वते योगं प्रोक्तवानहमव्ययम्
विवस्वान् मनवे प्राह मनुरिक्ष्वाकवेऽब्रवीत्

imaṁ vivasvate yogaṁ proktavān aham avyayam
vivasvān manave prāha manur ikṣhvākave ’bravīt

I taught this imperishable yoga to Vivasvan; Vivasvan taught it to Manu; and Manu to Ikshvaku.

Explanation

The chapter opens with a lineage.

Vivasvan to Manu to Ikshvaku — the sun, the first man, the founder of a royal line.

It is the Gita's only genealogy of itself, and it is a claim about transmission: this was not thought up, it was handed along. Which makes what 4.2 admits more striking.

Word by word

śhrī-bhagavān uvācha: the Supreme Lord Shree Krishna said
imam: this
vivasvate: to the Sun-god
yogam: the science of Yog
proktavān: taught
aham: I
avyayam: eternal
vivasvān: Sun-god
manave: to Manu, the original progenitor of humankind
prāha: told
manuḥ: Manu
ikṣhvākave: to Ikshvaku, first king of the Solar dynasty
abravīt: instructed