On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case

A method for constructing birational Darboux coordinates on a coadjoint orbit of the general linear group is presented. This method is based on the Gauss decomposition of a matrix in the product of an upper-triangular and a lower-triangular matrix. The method works uniformly for the orbits formed by the diagonalizable matrices of any size and for arbitrary dimensions of the eigenspaces.