What are the finite normal subgroups of $SO(4)$? If these do not exist (or if they are trivial, e.g. from some projection to $SO(2)$), are there different finite normal subgroups of $O(4),$ $U(4)$, or $SU(4)$?
[Many thanks to the StackExchange community for their help the last week or two as I piece together my understanding of a tough technical problem!]
The group $SO(4) = SO(4,\mathbb{R})$ does have a nontrivial finite normal subgroup of $\{\pm I \}$, and this in common with all even dimensions, because $\det(-I) = 1$. Geometrically this is reflection through the origin, and the Wikipedia article calls this central inversion. [Note that much of the technical content of this article seems to have been contributed by J.E. Mebius.]
The subgroup $\{\pm I \}$ is the centre of $SO(4)$, as it is in all even dimensions.
Unlike the higher even dimensions, $SO(4)$ has a couple of additional normal subgroups, although these are not finite. They are described in the article as the left-isoclinic rotations $S_L^3$ and the right-isoclinic rotations $S_R^3$. Each is isomorphic to the unit quaternions $S^3$.