suppose i have $O(3)$ as a group and then proceed to identify rotations on the same axis. That is, assuming an element in the simple component is written as
$ e^{s_i I_i } $
where $I_i$ are generators of rotations around axis X,Y and Z, basically the identification would work as
s_i \sim \lambda s'_i
for any real $\lambda$
questions:
What is this quotient group that i just obtained?
Are infinitesimal transformations in the obtained group still generated by $I_i$?