# Bott periodicity for Dirac matrices?

In most of the contexts I've seen, Bott periodicity refers to the periodicity of the homotopy groups of the manifolds of classical groups. However, recently I have heard of some physicist speaking of "Bott periodicity" as a periodicity in the reality properties of Dirac matrices, as summarized in Table 3 (page 8) in this document. However, it's not clear to me how the two kinds of periodicity are related. I'm looking for some explanations for this.