The low-degree real KO coefficient table

In the first four residues [F1] identifies the integer unit with the real line, the first order-two generator with the Möbius clutching and the second with the stable one-turn orthogonal clutching. Degree $-3$ is zero. Thus the first four entries are $\mathbb{Z},\mathbb{Z}\mathrm{/}2,\mathbb{Z}\mathrm{/}2,0$, including their geometrically specified generators.

The rank-four quaternionic spinor map in [F1] identifies $K{O}^{-4-j}(\mathrm{p}\mathrm{t})$ with $KS{p}^{-j}(\mathrm{p}\mathrm{t})$ for $0\le j\le 3$. The latter are $\mathbb{Z},0,0,0$, supplying exactly the remaining entries; the quaternionic line gives the integer generator. Multiplication by the actual rank-eight symbol then repeats these residues, so the example does not infer a real theorem from complex Bott or from a bare list of abstract groups.