Could you please show me an example of a principal $SO(3)$-bundle over $\mathbb{CP}^2$ such that the restriction of this bundle on a projective line is non-trivial.
Edit: One can try to construct the map $f\colon \mathbb{CP}^2 \mapsto \mathbb{CP}^1$ such that $f_{*}\colon H_2(\mathbb{CP}^2) \mapsto H_2(\mathbb{CP}^1)$ is an isomorphism.
On $\mathbb P^2(\mathbb C)$ you have the tautological complex line bundle $\xi$ .
The rank-3 real vector bundle $E=\xi\oplus \mathbb 1_\mathbb R$ can be reduced to an $SO_3(\mathbb R)$-vector bundle since it is orientable.
It has non trivial restriction to any complex line $\mathbb P^1(\mathbb C)\subset \mathbb P^2(\mathbb C)$ by the following calculation involving Chern and Stiefel-Whitney classes: $$w_2(E)=w_2(\xi)=c_1(\xi) \operatorname {mod} 2=\overline {-1}\ne \overline {0}\in H^2(\mathbb P^1(\mathbb C),\mathbb Z/2)=\mathbb Z/2$$