I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G_2$. However, when I do the same calculation for the quaternions, I end up with the three generators of $SO(3)$, which basically tells me that I can rotate the set of imaginary units anyway I like.
Intuitively, I don't understand, why it is not possible for the octonions to be rotated in the same way with an arbitrary rotation of $SO(7)$. Instead, the calculations show, that the possible transformations are restricted to the $G_2$ subgroup.
Is there way to understand this geometrically or algebraically? Is it related to the non-associative property of the octonions?
