  We study the homogeneous spaces of a simply connected, compact, simple Lie
group $G$ through the lens of K-theory. Our methods apply equally well to the
case where $G$ is in one of the four infinite families of classical groups, or
one of the five exceptional groups. The main examples we study in detail are
the four symmetric spaces FII, EIII, EVI, EVIII in Cartan's list of symmetric
spaces. These are, respectively, homogeneous spaces for $F_4$, $E_6$, $E_7$,
$E_8$ with dimensions $16$, $32$, $64$, $128$. They are the four Rosenfeld
projective planes.
