Transformation of a pair of commuting Hamiltonians quadratic in momenta to canonical form and real partial separation of variables for the Clebsch top

In the case of two degrees of freedom the simultaneous diagonalization of pairs of Hamiltonians quadratic on momenta that commute with respect to the standard Poisson bracket is considered. A general scheme of partial separation of variables for such pairs is discussed. As an example the Clebsch top is considered.