# Characteristic subgroups of $SO(4)$

lie-groupsorthogonal matrices

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 $$SO_4$$?

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.