Closed subgroups of $\mathrm{SO}(4)$

Question

My question is quite simple : we know all closed subgroups of $\mathrm{SO}(3)$; is it also known what are the closed subgroups of $\mathrm{SO}(4)$?

Answer 1

There is a complete list in the paper Mendes, Roberto de Maria Nunes. "Symmetries of spherical harmonics." Transactions of the American Mathematical Society 204 (1975): 161-178. As in the above answer this paper uses the 2 to 1 homomorphism $\phi:S^3 \times S^3 \rightarrow SO(4)$ . To summarize the construction in the paper for $G$ as subgroup of $SO(4)$ let L and R be the projections of $\phi^{-1}G$ on to the two components, these are closed subgroups of $S^3$ , and $L_K$ and $R_K$ are the normal subgroups $ (L,1) $ and $(1,R)$ of $L$ and $R$. We can then define an isomorphism $\Phi: L/L_K \rightarrow R/R_K$ by $\Phi(l L_K)= rR_k$ where $(l,r) \in \phi^{-1}G$.

Given $L,R, L_K,R_K,\Phi$ we can define $G$ by its action on a quaternion $q$ as $$ G = \{ g \in SO(4): g(q) = l q r^{-1} , l \in L, r \in R, \Phi(l L_K)= r R_K \} $$ All the closed subgroups of $SO(4)$ are enumerated as pairs $(L/L_K; R/R_K)$ a table in the paper.

Answer 2

There is an epimorphism $\mathrm{SU}(2) \times \mathrm{SU}(2) \to \mathrm{SO}(4)$ with the kernel $\langle(−I, −I)\rangle$. Since $\mathrm{SU}(2)$ is isomorphic to the unit quaternions, the epimorphism is given by $(u,v)\mapsto R_{u,v}$ where $R_{u,v}$ is the rotation of $\mathbb{R}^4$ given by $R_{u,v}(q)=v^{-1}qu$ for any quaternion $q$.

And $\mathrm{SU}(2)$ maps onto $\mathrm{SO}(3)$ with kernel $\langle -I\rangle$; again use quaternions.

As you say, you know the closed subgroups of $\mathrm{SO}(3)$, and so this gives the closed subgroups of $\mathrm{SU}(2)$ and so those of $\mathrm{SU}(2)\times \mathrm{SU}(2)$ (via Goursat's Lemma) and finally those of $\mathrm{SO}(4)$.

More generally, as it relates to semisimple subgroups, all simple subgroups of real Lie groups are known, as described here:

Karpelevič, F. I. The simple subalgebras of the real Lie algebras. Trudy Moskov. Mat. Obšč. 4 (1955), 3–112.

Karpelevič, F. I. Classification of the simple subalgebras of the real forms of classical algebras. Doklady Akad. Nauk SSSR (N.S.) 93, (1953). 613–616.

Karpelevič, F. I. Classification of the simple subgroups of the real forms of the group of complex unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 85, (1952). 1205–1208.