What is Ātman in the Bhagavad Gita?
Ātman is the self, but not in the sense of personality. Your temperament, memories and preferences all belong to the changing side of you. Ātman is what the Gita says is awake behind those and is not produced by them.
Chapter 2 makes the case in its most famous passage. Weapons do not cut it, fire does not burn it, water does not wet it. The argument is not that you are indestructible in any way you could test. It is that the thing being described is not the kind of thing those verbs apply to, the way a number cannot be set on fire.
15.7 adds the claim that most divides the commentators: this self is a fragment of the divine, eternal, entering the world of the living. Whether that fragment is ultimately identical with the whole, genuinely a part of it, or permanently distinct from it, is the oldest argument in Indian philosophy, and the Gita does not settle it. What all readings share is the practical point: what is awake in you did not begin at your birth and is not an output of your body.
The verses 19
2.12
There was never a time when I did not exist, nor you, nor these kings.
2.13
Just as the one who lives in this body passes through childhood, youth and old age, so it passes into another body.
2.16
What is not, never comes to be.
2.17
Know that what spreads through all of this cannot be destroyed.
2.18
These bodies end.
2.19
Whoever thinks this kills, and whoever thinks this is killed, have both misunderstood.
2.20
It is not born and it does not die.
2.22
As a person puts aside worn-out clothes and takes new ones, so the one who lives in a body puts aside worn-out bodies and goes into new ones.
2.23
Weapons do not cut it.
2.24
It cannot be cut, cannot be burned, cannot be soaked, cannot be dried.
2.29
One person looks at this and sees a wonder.
2.30
The one who lives in the body - in everyone's body - can never be killed.
6.5
Lift yourself by yourself.
6.6
For the one by whom the self has been conquered by the self, the self is a friend.
13.22
The person, seated in nature, experiences the strands born of nature.
13.31
When he sees that the separate states of beings all rest in one, and spread out from that one - at that point he reaches Brahman.
15.7
A fragment of my own self, lasting for ever, becomes the living being in the world of the living.
15.8
When the lord of the body takes a body up, and when he leaves one behind, he carries these with him, as the wind carries scents from where they sit.
15.9
Settled in hearing, in sight, in touch, in taste and in smell, and in the mind as well, he takes in the things of the world.