The Vedanta Sutras, commentary by Sankaracharya (SBE34), tr. by George Thibaut  at sacred-texts.com
13. The person within (the eye) (is Brahman) on account of the agreement (of the attributes of that person with the nature of Brahman).
Scripture says, 'He spoke: The person that is seen in the eye that is the Self. This is the immortal, the fearless, this is Brahman. Even though they drop melted butter or water on it (the eye) it runs away on both sides,' &c. (Kh. Up. IV, 15, 1).
The doubt here arises whether this passage refers to the reflected Self which resides in the eye, or to the individual Self, or to the Self of some deity which presides over the sense of sight, or to the Lord.
With reference to this doubt the pûrvapakshin argues as follows: What is meant (by the person in the eye) is the reflected Self, i.e. the image of a person (reflected in the eye of another): for of that it is well known that it is seen, and the clause, 'The person that is seen in the eye,' refers to it as something well known. Or else we may appropriately take the passage as referring to the individual Self. For the individual Self (cognitional Self, vigñânâtman) which perceives the colours by means of the eye is, on that account, in proximity to the eye; and, moreover, the word 'Self' (which occurs in the passage) favours this interpretation. Or else the passage is to be understood as referring to the soul animating the sun which assists the sense of sight; compare the passage (Bri. Up. V, 5, 2), 'He (the person in the sun) rests with his rays in him (the person in the right eye).' Moreover, qualities such as immortality and the like (which are ascribed to the subject of the scriptural passage) may somehow belong to individual deities. The Lord, on the other hand 1, cannot be meant, because a particular locality is spoken of.
Against this we remark that the highest Lord only can be meant here by the person within the eye.--Why?--'On account of the agreement.' For the qualities mentioned in the passage accord with the nature of the highest Lord. The quality of being the Self, in the first place, belongs to the highest Lord in its primary (non-figurative or non-derived) sense, as we know from such texts as 'That
is the Self,' 'That art thou.' Immortality and fearlessness again are often ascribed to him in Scripture. The location in the eye also is in consonance with the nature of the highest Lord. For just as the highest Lord whom Scripture declares to be free from all evil is not stained by any imperfections, so the station of the eye also is declared to be free from all stain, as we see from the passage, 'Even though they drop melted butter or water on it it runs away on both sides.' The statement, moreover, that he possesses the qualities of samyadvâma, &c. can be reconciled with the highest Lord only (Kh. Up. IV, 15, 2, 'They call him Samyadvâma, for all blessings (vâma) go towards him (samyanti). He is also vâmanî, for he leads (nayati) all blessings (vâma). He is also Bhâmanî, for he shines (bhâti) in all worlds'). Therefore, on account of agreement, the person within the eye is the highest Lord.
124:1 Who, somebody might say, is to be understood here, because immortality and similar qualities belong to him not somehow only, but in their true sense.