This is too long for a comment.

I found the article 

<cite authors="Lazaroiu, Calin Iuliu; Babalic, Elena Mirela; Coman, Ioana Alexandra">_Lazaroiu, Calin Iuliu; Babalic, Elena Mirela; Coman, Ioana Alexandra_, [**Geometric algebra techniques in flux compactifications**](http://dx.doi.org/10.1155/2016/7292534), Adv. High Energy Phys. 2016, Article ID 7292534, 42 p. (2016). [ZBL1366.83098](https://zbmath.org/?q=an:1366.83098).</cite>

on the classification of Killing (s)pinors using geometric algebra with applications to $\mathcal N=1$ M-theory compactifications to 3D.

Their perspective on geometric algebra is explained in section 3. The central object seems to be what they call the Kaehler-Atiyah algebra over some (pseudo)Riemannian manifold $M$ which as far as I can tell will reduce to geometric algebra as in Wikipedia when $M$ is Minkowski space. They also sketch how the KA algebra is obtained by a quantization procedure.