Quadratic Hamiltonians with a linear Lie-Poisson bracket have a number of applications in mechanics. For example, the Lie-Poisson bracket e(3) includes the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and the Kirchhoff model describes the motion of a rigid body in ideal fluid. Among the applications with a Lie-Poisson bracket so(4) and so(3, 1) is the description of free rigid body motion in a space of constant curvature. Advances in computer algebra algorithms, in implementations and hardware, together allow the computation of Hamiltonians with higher degree first integrals providing new results in the classic search for integrable models. A computer algebra module enabling related computations in a 3-dimensional vector formalism is described.