  In this paper we analyze the normal forms of a general quadratic Hamiltonian
system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ -
skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric
matrix. A consequence of the main results is that any first-order autonomous
three-dimensional differential equation possessing two independent quadratic
constants of motion which admits a positive/negative definite linear
combination, is affinely equivalent to the classical "relaxed" free rigid body
dynamics with linear controls.
