Given a vector space, $V$, with a quadratic form $Q$, one possible definition for its Clifford algebra (and perhaps the one I like the most) is to take it to be the quotient of the tensor algebra of $V$, $T(V)$, modulo the ideal $\langle{u\otimes{u}-Q(u)\bf{1}}\rangle$.
Now, I have tried to track this construction back to papers/books, but I haven't been really successful. The oldest reference I found where this is used is the famous article "Clifford modules" by Atiyah, Bott and Shapiro published in 1964. However, in the introduction they mention that the section in which this definition is stated «contains nothing essentially new, though we formulate the results in a novel way», thus I am not sure if the definition itself is "formulated in a novel way" or "nothing new". Of all the references in that article that I was able to find, none of them use this definition.
I am very curious to know if that was the first time this construction was used, or if it had been used before the publication of this article.