Given angle $\theta$ and a subspace $P\leq M_4(\mathbb{R})$ of dimension 2 (a plane), how do I compute for the simple rotation matrix $M$ of angle $\theta$ that fixes $P$?
I tried finding the entries of $M$ using the images of basis elements of $P$ but it leads nowhere. Also, the example in this wikipedia page seems to be very specific so I can't relate it to general cases.
Take $\{e_1, e_2\}$ as an orthonormal basis of $P$. Complement it into a basis $\{e_1, e_2, e_3, e_4\}$ of $\mathbb R^4$. In that basis, the matrix $M$ is
$$M=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & \cos \theta & - \sin \theta\\ 0 & 0 & \sin \theta & \cos \theta \end{pmatrix}$$
With this, all points of $P$ are fixed.