Unit or non-zero octonions form an $A_\infty$-space?

Question

If I have a Moufang loop, can it have a classifying space? I'm thinking of the unit octonions, if that's too general, so perhaps a better question is: are the unit (or non-zero) octonions an $A_\infty$-space? I've had a poke around, and this seems like the sort of thing that should be known.

Answer 1

No. If unit octonions were $A_\infty$-space, $BS^7$ («$OP^\infty$») would be a space with cohomology ring $\mathbb Z[x]$, $\deg x=8$ (this follows from LHSS for Serre fibration $\text{pt}\to\Omega BS^7\cong S^7\to BS^7$) — which is impossible (see e.g. Corollary 4L.10 if Hatcher's «Algebraic topology»).

P.S. In fact, there are no even just homotopy associative multiplications on $S^7$ — see James. Multiplication on Spheres (II). Trans. AMS, vol. 84, no 2 (1957), pp. 545–558.