In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie Groups. I am not precisely sure what I am looking for, but ostensibly there should be answers forth coming from other who have promised such answers. I understand a bit about how the exceptional Lie groups come up historically, and please correct the following if it is incorrect, but when looking at the possible dynkin diagrams you see that there is no reason for $E_6$,$E_7$,$E_8$,$G_2$, and $F_4$ to not occur as root systems. While root systems are geometric, this is not what I am asking about.

Thanks