In his lecture Thoughts on the Gita, Swami Vivekananda says the following. I'm interested in the part that's in bold.
THOUGHTS ON THE GITA
The book known as the Gita forms a part of the Mahābhārata. To understand the Gita properly, several things are very important to know. First, whether it formed a part of the Mahabharata, i.e. whether the authorship attributed to Veda-Vyāsa was true, or if it was merely interpolated within the great epic; secondly, whether there was any historical personality of the name of Krishna; thirdly, whether the great war of Kurukshetra as mentioned in the Gita actually took place; and fourthly, whether Arjuna and others were real historical persons.
Now in the first place, let us see what grounds there are for such inquiry. We know that there were many who went by the name of Veda-Vyasa; and among them who was the real author of the Gita — the Bādarāyaṇa Vyasa or Dvaipāyana Vyasa? "Vyasa" was only a title. Anyone who composed a new Purāna was known by the name of Vyasa, like the word Vikramāditya, which was also a general name. Another point is, the book, Gita, had not been much known to the generality of people before Shankarāchārya made it famous by writing his great commentary on it. Long before that, there was current, according to many, the commentary on it by Bodhāyana. If this could be proved, it would go a long way, no doubt, to establish the antiquity of the Gita and the authorship of Vyasa. But the Bodhayana Bhāshya on the Vedānta Sutras — from which Rāmānuja compiled his Shri-Bhāshya, which Shankaracharya mentions and even quotes in part here and there in his own commentary, and which was so greatly discussed by the Swami Dayānanda — not a copy even of that Bodhayana Bhashya could I find while travelling throughout India. It is said that even Ramanuja compiled his Bhashya from a worm-eaten manuscript which he happened to find. When even this great Bodhayana Bhashya on the Vedanta-Sutras is so much enshrouded in the darkness of uncertainty, it is simply useless to try to establish the existence of the Bodhayana Bhashya on the Gita. Some infer that Shankarāchārya was the author of the Gita, and that it was he who foisted it into the body of the Mahābhārata.
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Who were these people that thought Śaṅkarācārya was the author of Bhagavad-gītā?
