@EricPeterson My problem with this reference is that it does not show that the map ΣKO→KO is multiplication by eta, which for me is a huge part of the Wood fiber sequence. I guess one can deduce it from some computation of KO^*CP^2, but it would be nice to know if Wood actually stated this anywhere...

totally. like i said, i haven’t read the paper, but i’d readily believe that it’s missing that. dunno if there’s another wood reference that might include it — i don’t see anything promising — & i don’t know the oldest identification of the map.