The simple classical compact groups SO(n), SU(n), Sp(n) are related
to the division algebras R,C,H (reals, complex numbers, quaternions).
The exceptional groups G_2, F_4, E_6, E_7, E_8 are somehow related
to the remaining normed division algebra, the octonions O, but
the relation is not quite easy to understand, except for G_2 = Aut(O).
We will start with Hurwitz' theorem that R,C,H,O are all possible
normed division algebras over the reals (including construction).
Then we will turn to the octonionic projective plane and its
connection to F_4 and E_6. I might tell what I know about
the relation to E_7 and E_8. If there is interest, I could
also explain the relation of the octonions to real Bott