I'm interested in the non-discrete Lie subgroups. (I've searched long for a resource that lists things like all normal subgroups for group a $G$, but have not found one.)
Background
My understanding (with $\triangleleft$ meaning normal subgroup):
$SU(2)\triangleleft SO(4)$ because isoclinic rotations in $SO(4)$ are normal (Wikipedia) and isomorphic to $SU(2)$ (answer here)
$SO(3)\triangleleft SO(4)$ because $SO(4)/SO(3)=S^3\cong SU(2)$, and because $SU(2)$ is a Lie group, and in $G/N=H$, $N$ is normal iff $H$ is a Lie group (rather than a non-group homogenous space) according to the answer here. (EDIT: per Jason DeVito's comment below, my reasoning here was wrong: the quotient is $S^3$ without the structure of $SU(2)$.)
Question: are there other non-discrete $N\triangleleft SO(4)$?