Rotation in 4D space I want to find the rotaion matrix in 4D space for rotating a vector in the plane which is perpendicular to the vectors $(1,1,1,1)$ and $(1,1,1,0)$. I know how to do in 3D space. I can't make visualise the procedure in 4D space.
 A: Here is one procedure.
Step 1: Obtain an orthonormal basis for the plane of rotation. In this case, $\{\frac 1{\sqrt{3}}(1,1,1,0),(0,0,0,1)\}$ is such a basis.
Note: The orientation of this basis matters! To switch the orientation, you could switch the order of the two vectors or invert the sign of all entries of one of the vectors.
Step 2: Extend this to an orthonormal basis of $\Bbb R^4$. In this case, we could take
$$
\mathcal B = \left\{\frac 1{\sqrt{3}}(1,1,1,0),(0,0,0,1),\frac 1{\sqrt{2}}(1,-1,0,0),\frac 1{\sqrt{6}}(1,1,-2,0)\right\}.
$$
Step 3: Let $U$ denote the matrix whose columns are the elements of $B$. Let $R$ denote the rotation (in the $x_1x_2$ plane) by the desired angle. That is,
$$
R = \pmatrix{\cos \theta & -\sin \theta & 0&0\\
\sin\theta & \cos\theta & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1}.
$$
Then the rotation matrix that you're looking for is given by $URU^T$.

Note: If you want to make computation marginally more efficient, you could also express the matrix as follows. Let $u_1,u_2,u_3,u_4$ denote the (column-)vectors from the basis $\mathcal B$, i.e. the columns of $U$. Then
$$
URU^T = \pmatrix{v_1 & v_2} \pmatrix{\cos \theta & - \sin \theta\\ \sin \theta & \cos \theta} \pmatrix{v_1 & v_2}^T + \pmatrix{v_3 & v_4}\pmatrix{v_3 & v_4}^T.
$$
