Inquiry is careful, first-person looking at your own experience. Its aim is to know the truth of the teaching directly, rather than take it on trust.
You take the question at hand and look at what is actually there, slowly and honestly. You do not rush to a conclusion. You do not borrow one. You stay with the question until something becomes clear from your own looking.
The bucket exercise you did earlier was inquiry. You did not just read about awareness as a theory. You sorted body, thoughts, emotions, and self-image into the not-me bucket. You found it impossible to put anything into the me bucket. Then you knew directly, not as an idea but in your own immediate experience, the awareness that let you experience everything else. All of this was your own finding.
That is what real inquiry looks like: first-person, slow, careful, aimed at your own direct knowing rather than at someone else's conclusion.
Inquiry is different from study, and the two are worth distinguishing. Study is reading books, listening to talks, picking up the ideas and vocabulary of a tradition. It is collecting information about what is real. Study is useful. It introduces the questions, sketches the landscape, and gives you the language to think about what you are looking at. Without some study, you would not even know what to inquire into.
But study is not inquiry. The ideas you pick up through study are second-hand. Someone else did the looking. You are reading their report. You may agree with the report. You may understand it well. But agreeing with a true conclusion is not the same as knowing directly what it points to.
Self-knowledge is what frees you. And self-knowledge is not the idea of who you are. It is the direct knowledge of what you actually are. A borrowed conclusion, however well stated, cannot free you. Only your own direct knowing can.
The Gita is not trying to produce clever students who can argue well. It is trying to produce students who know directly. That direct knowing requires inquiry.
The bucket inquiry is a good example. Let's take another: the inquiry into doership.
Something small happened today. You said something to someone — a sharp word, a kind word, just a comment in a meeting. Bring it to mind.
Replay it slowly.
Watch the urge to speak arise, before any words came out. Did you choose to have that urge, or did it just appear? Watch the body move — the breath, the lips, the tongue. Did you choose each muscle, or did the body act on its own once the urge was there? Watch the moment afterward when you said to yourself, I did this.
Sit with the question. You are not trying to come up with a clever answer. You are trying to look at something you usually skip past — the assumption that you produce your actions, the way a hand makes a fist. Inquiry slows the looking down enough that the assumption can be examined.
Notice that this inquiry and the bucket inquiry point in the same direction.
The bucket inquiry showed you that what you are is the awareness in which body and mind appear — not the body and mind themselves. Even trying your best, you could not find a knower. The action inquiry shows you that the doer — the "I" who supposedly produces the action — cannot be found either. Same misidentification. Two angles on the same recognition.