SO(4)

Overview

Since a reflection is orientation-reversing, any isometry that preservers orientation (rotation) must be product of even number of reflections. (by Cartan–Dieudonn´e theorem)

Furthermore, we see that a product of two reflections gives a planar rotation, also called a simple rotation. Therefore, any “high dimensional rotation” (formally orientation-preserving orthogonal map) is a product of several (orthogonal) two dimensional rotations

More Specifically, in ${\mathbb{R}}^4$, any rotation is a product of 0, 2 or 4 reflections or two rotations. Since each reflection can be represented by a unit quaternion orthogonal to the hyperplane of reflection, by intuition we would need 4 quaternions. However, it is also possible to represent it with two quaternions.

representation of reflections (quaternion)

let $u$ be the unit quaternion orthogonal to the hyperplane of reflection $q\mapsto -u\bar{q}u$ sends $q$ to its reflection

Why?

Note that $q\mapsto -\bar{q}$ reverses the real part of $q$ and it sends $vu$ to $-\bar{v}u$ This means $u$ is sent to $-u$ while $iu,ju,ku$ (rotate orthogonally in $\mathbb{H}$) are preserved Furthermore left/right multiplication by an unit quaternion is an isometry

representation of rotations (quaternion)

Combining multiple reflections we get

$$q\mapsto u_{2n}\dots {\bar{u}}_3{u}_2{\bar{u}}_{1}q{\bar{u}}_1{u}_2{\bar{u}}_3\dots u_{2n}$$

which can be simplified to form $q\mapsto vqw$ where $v,w$ are unit quaternions

Relation to other groups

The map from $SU(2)\times SU(2)$ to $SO(4)$

There exists a 2-to-1-homomorphism from $SU(2)\times SU(2)$ to $SO(4)$

So,

$$SO(4)\cong \frac{SU(2)\times SU(2)}{\{\pm (1,1)\}}$$

$SU(2)$ represents the group of unit quaternions (see Subgroup) This makes sense because rotation in ${\mathbb{R}}^4$ can be represented as two quaternions.

Simplicity

$SO(4)$ is not simple