A hyperkähler metric on twisted cotangent bundles of the complex projective space

We construct a hyperkähler metric on twisted cotangent bundles of the complex projective space $\mathbb{CP}^n$ explicitly in terms of local coordinates. Note that the twisted cotangent bundles of $\mathbb{CP}^n$ are holomorphically isomorphic to complex semisimple coadjoint orbits of $\mathrm{SL}_{n+1}(\mathbb{C})$.