The Vedanta Sutras, commentary by Sankaracharya (SBE34), tr. by George Thibaut  at sacred-texts.com
17. (The person within the eye is the highest), not any other Self; on account of the non-permanency (of the other Selfs) and on account of the impossibility (of the qualities of the person in the eye being ascribed to the other Selfs).
To the assertion made in the pûrvapaksha that the person in the eye is either the reflected Self or the cognitional Self (the individual soul) or the Self of some deity the following answer is given.--No other Self such as, for instance, the reflected Self can be assumed here, on account of non-permanency.--The reflected Self, in the first place, does not permanently abide in the eye. For when some person approaches the eye the reflection of that person is seen in the eye, but when the person moves away the reflection is seen no longer. The passage 'That person within the eye' must, moreover, be held, on the ground of proximity, to intimate that the person seen in a man's own eye is the object of (that man's) devout meditation (and not the reflected image of his own person which he may see in the eye of another man). [Let, then, another man approach the devout man, and let the latter meditate on the image reflected in his own eye, but seen by the other man only. No, we reply, for] we have no right to make the (complicated) assumption that the devout man is, at the time of devotion, to bring close to his eye another man in order to produce a reflected image in his own eye. Scripture, moreover, (viz. Kh. Up. VIII, 9, 1, 'It (the reflected Self) perishes as soon as the body perishes,') declares the non-permanency of the reflected Self.--And, further, 'on account of impossibility' (the person in the eye cannot be the reflected Self). For immortality and the other qualities ascribed to the person in the eye are not to be perceived in the reflected Self.--Of the cognitional Self, in the second place, which is in general connexion with the whole body and all the senses, it can likewise not be said that it has its permanent station in the eye only. That, on the other hand, Brahman although all-pervading may, for the purpose of contemplation, be
spoken of as connected with particular places such as the heart and the like, we have seen already. The cognitional Self shares (with the reflected Self) the impossibility of having the qualities of immortality and so on attributed to it. Although the cognitional Self is in reality not different from the highest Self, still there are fictitiously ascribed to it (adhyâropita) the effects of nescience, desire and works, viz. mortality and fear; so that neither immortality nor fearlessness belongs to it. The qualities of being the samyadvâma, &c. also cannot properly be ascribed to the cognitional Self, which is not distinguished by lordly power (aisvarya).--In the third place, although the Self of a deity (viz. the sun) has its station in the eye--according to the scriptural passage, 'He rests with his rays in him'--still Selfhood cannot be ascribed to the sun, on account of his externality (parâgrûpatva). Immortality, &c. also cannot be predicated of him, as Scripture speaks of his origin and his dissolution. For the (so-called) deathlessness of the gods only means their (comparatively) long existence. And their lordly power also is based on the highest Lord and does not naturally belong to them; as the mantra declares, 'From terror of it (Brahman) the wind blows, from terror the sun rises; from terror of it Agni and Indra, yea, Death runs as the fifth.'--Hence the person in the eye must be viewed as the highest Lord only. In the case of this explanation being adopted the mention (of the person in the eye) as something well known and established, which is contained in the words 'is seen' (in the phrase 'the person that is seen in the eye'), has to be taken as referring to (the mental perception founded on) the sâstra which belongs to those who know; and the glorification (of devout meditation) has to be understood as its purpose.