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 ${\mathbb{C}\mathbb{P}}^n$ explicitly in terms of local coordinates. Note that the twisted cotangent bundles of ${\mathbb{C}\mathbb{P}}^n$ are holomorphically isomorphic to complex semisimple coadjoint orbits of ${\mathrm{S}\mathrm{L}}_{n+1}(\mathbb{C})$.