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A key point we want to stress is that the 10-fold way is usually viewed as 10 = 2+8, where 2 and 8 are the periodicities in complex and real K-theory. And then the K-theory classification of topological phases is criticized because it only applies to free systems. However, we believe this viewpoint is slightly misguided. The unifying concept is really that of a real super-division algebra, and there are 10 such. They can be parceled into 10 = 8+2 but they can also equally naturally be parceled into 10 = 7+3 (with the 3 referring to the purely even superdivision algebras).
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Moore argues (Moore 13, p. 129) that the proper basis of the 10-fold way is the classification of real super-division algebras, and conjectures a 1-1 correspondence between these and Dyson’s 10 types (p. 132).
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