at.algebraic topology - Relation between $KO$ and $K$

Every real bundle can be complexified so there's a natural transformation $KO(X) to K(X)$. Going all the way around multiplies by $2$ each time so if you localise at the prime $2$, you get a nice splitting.

More generally, these fit into a long exact sequence. I don't remember if the third term has a special name or not. (As this is a community wiki question, someone should feel free to edit this answer to add it in.) I've seen its classifying space written as $B(U/O)$ which suggests not.