There are exactly 3 nontrivial proper normal subgroups of $ SO_4 $. This can be seen by using Goursat's lemma on the double cover $ SU_2 \times SU_2 $ and then noting that only 5 of the 10 normal subgroups in $ SU_2 \times SU_2 $ descend to the quotient $ SO_4 $ (in other words only 5 contain the element $ (-1,-1) $ ). Of these five normal subgroups, two are just the trivial group and all of $ SO_4 $. The remaining groups are the center $ \pm I $ and the groups of left and right isoclinic rotations, respectively, both of which are isomorphic to $ SU_2 $. Are these two copies of $ SU_2 $ characteristic subgroups? (If not then the outer automorphism given by conjugation by a reflection in $ O_4 $ must switch the two copies of $ SU_2 $.)

Question: Are the groups of left and right isoclinic rotations characteristic? Or can they be exchanged by the outer automorphism of $ SO_4 $?

## Best Answer

A left isoclinic rotation is of the form $$ \begin{bmatrix}a&-b&-c&-d \\ b&a&-d&c\\c&d&a&-b\\d&-c&b&a\end{bmatrix}$$ where $a,b,c,d\in\mathbb{R}$ such that $a^2+b^2+c^2+d^2=1 $. These matrices form an $ SU(2) $ subgroup known as the group of left isoclinic rotations.

A right isoclinic rotation is of the form $$ \begin{bmatrix} a&-b&-c&-d \\ b&a&-b&-c\\ c&b&a&d\\ d&c&-d&a \end{bmatrix} $$ where $a,b,c,d\in\mathbb{R}$ such that $a^2+b^2+c^2+d^2=1$. These matrices form an $ SU(2) $ subgroup known as the group of right isoclinic rotations.

These two normal subgroups groups are not characteristic. The outer automorphism of $ SO(4) $ given by conjugation by the reflection $$ \begin{bmatrix} 1&0&0&0 \\ 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0 \end{bmatrix} \in O(4) $$ interchanges the group of left isoclinic rotations and the group of right isoclinic rotations.