Rotation in 4 dimensions around an arbitrary plane

Question

Rotations in 4 dimensions are performed around a fixed plane, they can be described by $SO(4)$, which is a group of orthogonal matrices with determinant equal to 1. It is easy to derive rotation matrices around the coordinate planes in $\mathbb{R}^4$, for example, $$ \begin{pmatrix} \cos(\theta) & \sin(\theta) & 0 & 0 \\ -\sin(\theta) & \cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ performs a rotation around $X_3X_4$-plane by an angle $\theta$. What I am interested is are rotations around an arbitrary plane in $\mathbb{R}^4$. Given such a plane, how to get a rotation matrix for this plane?

Answer 1

Let $v,w$ be an arbitrary orthonormal pair of vectors in that plane. Extend it to an orthonormal basis of $\Bbb R^4$. Then the matrix that you gave above is a rotation matrix in terms of that basis.

To express it in another basis, just perform a change of basis to your desired basis.