In a previous question by me I asked about Berger spheres and their Lorentzian analogue, squashed $AdS_3$ along Hopf fibres. It was well answered (by https://mathoverflow.net/users/13268/ben-mckay) but now I try to find literature about Berger spheres and their construction as in the answer, with very little success. For completeness I post how Berger spheres are constructed in the mentioned answer:
It is easier to simply write that the Hopf fibration is given by taking any connected 1-dimensional subgroup $K$ of $S^3$, and then the quotient $S^3 \to S^3/K$ is a Hopf fibration. Since the adjoint action of $S^3$ acts transitively on 1-dimensional subspaces of its Lie algebra (i.e. rotations of 3-dimensional Euclidean space act transitively on lines through the origin), $S^3$ acts transitively, by conjugation, on all connected 1-dimensional Lie subgroups. Hence the choice of $K$ is arbitrary. In your basis $z_1$, $z_2$,$z_3$ for the Lie algebra of $S^3$, you could just take dual basis, say $\omega_1$,$\omega_2$,$\omega_3$, and then your Hopf fibration can have fibers given by setting two linearly independent linear combinations of these two zero, say $0=\omega_1=\omega_2$, so the subgroup $K$ is then tangent to $z_3$. We can make the Berger metric then be anything of the form $\omega_1^2+\omega_2^2+ \lambda \omega_3^2$, carried around $S^3$ by left (or right, if you prefer) translation.