This practice recurs in several traditions and is expounded upon by multiple advocates, but is originally conceived of by Sri Ramana Maharshi. The practice is not always called “self-inquiry”; seekers should be alert to correspondences found in other writings. Maharshi remains the principal proponent even 70 years after his death, and his works, including the pamphlet “Who Am I?”, remain the most concise and descriptive expositions of the meditation. What is presented here is only a sampling.

9. What is the path of inquiry for understanding the nature of the mind? That which rises as ‘I’ in this body is the mind. If one inquires as to where in the body the thought ‘I’ rises first, one would discover that it rises in the heart. That is the place of the mind’s origin. Even if one thinks constantly ‘I’ ‘I’, one will be led to that place. Of all the thoughts that arise in the mind, the ‘I’ thought is the first. It is only after the rise of this that the other thoughts arise. It is after the appearance of the first personal pronoun that the second and third personal pronouns appear; without the first personal pronoun there will not be the second and third.

10. How will the mind become quiescent? By the inquiry ‘Who am I?’. The thought ‘who am I?’ will destroy all other thoughts, and like the stick used for stirring the burning pyre, it will itself in the end get destroyed. Then, there will arise Self-realization.

11. What is the means for constantly holding on to the thought ‘Who am I?’ When other thoughts arise, one should not pursue them, but should inquire: ‘To whom do they arise?’ It does not matter how many thoughts arise. As each thought arises, one should inquire with diligence, “To whom has this thought arisen?”. The answer that would emerge would be “To me”. Thereupon if one inquires “Who am I?”, the mind will go back to its source; and the thought that arose will become quiescent. With repeated practice in this manner, the mind will develop the skill to stay in its source. When the mind that is subtle goes out through the brain and the senseorgans, the gross names and forms appear; when it stays in the heart, the names and forms disappear. Not letting the mind go out, but retaining it in the Heart is what is called “inwardness” (antarmukha). Letting the mind go out of the Heart is known as “externalisation” (bahir-mukha). Thus, when the mind stays in the Heart, the ‘I’ which is the source of all thoughts will go, and the Self which ever exists will shine. Whatever one does, one should do without the egoity “I”. If one acts in that way, all will appear as of the nature of Siva (God).

Sri Ramana Maharshi, “Who Am I?”

Merrell-Wolff also stumbled upon a variation of this practice, as discussed in Pathways though to Space and his personal correspondence.

The form of meditation which I have found effective differs substantially from that generally given in the manuals of meditation. Repeatedly, I have tried stopping thought and closing out the senses, but the artificial state thus effected was barren of results….Now, within a process or manifold, a given phase or aspect may be isolated for special attention without stopping the process or eliminating the balance of the manifold…When I recalled this fact and applied it, I found at once a really effective method of meditation. It was by applying this method of isolation of the essential element in the midst of a complex, without trying to restrain the other components, that the Transition was effected during the early part of this month.

Franklin Merrell-Wolff, II. Concerning Meditative Technique, Pathways Through to Space

I abstracted the subjective moment — the “I AM” or “Atman” element — from the totality of the objective consciousness manifold. Upon this I focused. Naturally, I found what, from the relative point of view, is Darkness and Emptiness. But I Realized It as Absolute Light and Fullness and that I was That.

Franklin Merrell-Wolff, I. The Light Breaks Forth, Pathways Through to Space

Now, I know that Self Realization is possible, but this is not knowledge of an object. Perhaps we might call the process an inverse cognition; I shall have to describe what I mean by this. If one studies the process of cognition, either sensual or conceptual, with careful subtlety he will find something like a flow out toward the object. This flow may be likened to a light-ray. The flow can be observed, itself. In some measure and it can be more or less completely stopped.

The object can be made to disappear and in its place may be found either a sense of darkness or of light. It may even induce an ecstatic state of more or less intensity. Now reverse the flow, which is a process of profound introversion, and you have Self-realization. It is a state of the Light centered in Itself and not flowing to objects. It is like beginning a judgment starting with “I” and going no further. This is the isolation of the subjective-moment. The absolute dissolution of the object is not necessary, for one may achieve his realization by reflecting only part of the ray back. This avoids trance. 

From ‘A Letter to Reid Gardner’ (1949) on the website of The Franklin Merrell-Wolff Fellowship.