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 $I_i$?
Every rotation can be gotten as a conjugate of a rotation around, say, the $x$ axis, so the only way to kill, say, every $(\lambda-1)s_x$ with a group homomorphism is to map all of $SO(3)$ to the identity.
Note that the rotations around one axis form a subgroup, but not a normal subgroup.
If you don't care about the group structure but only the topology of SO(3), and exclude the identity before you start quotienting, you get the real projective plane.