I'm watching John Carlos Baez (npub1nf4…nqe4) 's very nice talk https://www.youtube.com/watch?v=ulIeYBgBtaQ and the question occurs to me: instead of artificially looking at the connected component of the stabiliser of the copy of h_3(C) inside h_3(O), and then intersect that with the stabiliser of the h_2(C) inside that, can we instead take the intersection of three stabiliser subgroups, and the third one being the stabiliser of "a copy of C", because the non-identity component does complex conjugation (in some sense). However, "h_1(C)" is, if I'm not mistaken, isomorphic to the reals (as it's hermitian 1x1 complex matrices which just contain a single real entry). And the 1+1-dimensional spin factor is not isomorphic to C as an algebra. So it's unclear what's happening. One might imagine perhaps that it's something like "the stabiliser of a complex structure on h_3(O)", where I'm not even convinced such a thing exists.
And I'm sure that John and Paul Schwahn (npub1zz2…9mhx) will almost surely have tried to think about such a thing already, but I don't think I've seen it mentioned (in the things I've read, perhaps I missed something).
