Attributed to Adi Shankara
«Brahman is real, the world is illusory; the individual soul is none other than brahman.»
Misattributed
Verdict
It is not in Adi Shankara's work
It is the summary of Advaita most often quoted under Śaṅkara's name, but it does not appear in the authentic commentaries. The first half-line is in Vivekacūḍāmaṇi 20 (attributed); the full formula, with "jīvo brahmaiva nāparaḥ", circulates as a verse of the Brahmajñānāvalīmālā, another traditionally ascribed work. It is a school summary faithful in essence, not a quotation from the author of the bhāṣyas.
What they did say
Verified words of Adi Shankara
«It is a matter not requiring any proof that the object and the subject whose respective spheres are the notion of the 'Thou' (the Non-Ego) and the 'Ego', and which are opposed to each other as much as darkness and light are, cannot be identified.»
Brahmasūtrabhāṣya, prólogo sobre la superposición (adhyāsabhāṣya), frase inicial; trad. inglesa de Thibaut, SBE 34 (1890); versión española del investigador
«Superimposition is the apparent presentation, in the form of remembrance, to consciousness of something previously observed, in some other thing.»
Brahmasūtrabhāṣya, adhyāsabhāṣya (definición de adhyāsa); Thibaut, SBE 34
«For everybody is conscious of the existence of (his) Self, and never thinks 'I am not'.»
Brahmasūtrabhāṣya 1.1.1; Thibaut, SBE 34