What is Brahman in the Bhagavad Gita?
Brahman is the whole, the single reality that everything else is a form of. The word comes from a root meaning to grow or swell, and it names what the Upanishads were pointing at before the Gita took the question up.
The Gita's contribution is to put Brahman in contact with ordinary life rather than leaving it as an abstraction. 5.10 says a person who acts while offering everything to Brahman is untouched by what he does, like a lotus leaf in water. 18.54 describes becoming Brahman as a condition you can recognise from outside: not grieving, not wanting, the same towards everything alive.
The detail most readers miss is where that verse goes next. Having become Brahman, the person attains the highest devotion. So the calm, unmoved state is not the destination in this book. It is the ground that clears so something warmer can begin. That ordering is the Gita's particular answer to a question the older texts had left in a cooler place.
The verses 13
4.24
The offering is the infinite.
5.10
Whoever acts placing his actions in the infinite, having given up attachment, is not stained by wrong - as a lotus leaf is not wetted by water.
5.19
Even here, coming and going has been conquered by those whose minds stand in evenness.
5.20
He does not leap up when something pleasant comes, and does not recoil when something unpleasant does.
5.21
With his self unattached to outside contacts, he finds the happiness that is in the self.
8.3
Krishna said: The imperishable, the highest, is Brahman.
13.12
… constancy in the knowledge of the self, and keeping in view what the knowledge of truth is for.
13.30
Whoever sees that actions are performed in every way by nature alone, and that the self is not the doer - that person sees.
14.26
And whoever serves me with unwavering devotion, and nothing else, goes beyond these strands and becomes fit to be Brahman.
14.27
For I am the ground of Brahman - of what does not die, of what does not wear out, of the lasting way, and of a happiness that has no end.
18.50
Learn from me, briefly, how someone who has reached that fulfilment also reaches the absolute, which is the highest ground that knowing can stand on.
18.53
Freed from self-importance, force, swagger, wanting, anger and holding on to things, without any sense of mine, at peace: such a person is ready to become the absolute.
18.54
Having become the absolute, clear in himself, he does not grieve and does not want.