When we use the polar coordinates on $\mathbb{R}^4$, which are \begin{aligned} &t\\ &x=r \sin \theta \cos \phi \\ &y=r \sin \theta \sin \phi \\ &z=r \cos \theta, \end{aligned}
the sphere symmetric metric is \begin{aligned} \mathrm{d} s^{2}=-A(r, t) d t^{2}+B(r, t) d t d r+C(r, t) d r^{2}+D(r, t)\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right) \end{aligned}
where $A,B,C,D$ is a function depending on $r,t$.
Here, the term $\sin ^{2} \theta \mathrm{d} \phi^{2}$ looks like depending on the angle $\theta$ and it looks like a contradiction to the sphere symmetry.
I think that the sphere symmetric metric does not depend on the angles $\theta,\phi$ but depends only on the radius $r$. Is it a mistake? Please, teach me the definition of sphere symmetry.