I have an integral at hand which has the form of
$I = \int_{u\in \mathbb{S}^2} f(\mathbf{u}\cdot \mathbf{s}_1) f(\mathbf{u}\cdot \mathbf{s}_2) d\mathbf{u}$ where $\mathbb{S}^2$ is the unit sphere with radius $1$ and $\mathbf{u}\cdot \mathbf{s}_1$ is the inner product of the two vectors.
Intuitively this integration should depend only on the relative position (say, the spherical angle) between $\mathbf{s}_1$ and $\mathbf{s}_2$ since $\mathbf{u}$ is integrated over the whole sphere surface. My question is how to prove $I$ only depend on the spherical angle between $\mathbf{s}_1$ and $\mathbf{s}_2$ without knowing the explicit form the $f$. I am thinking about rotating $\mathbf{u}$ but cannot write an explicit formula since the spherical angle has a complicated form in terms of the spherical coordinates.