  The stability of a special class of equilibria for the free rigid body on
$\mathfrak{so}(5)$ is discussed. An instability region and two stability
regions are established. The list of constants of motion which assure the
complete integrability of the system is presented and used in the approach of
stability with energy methods. The importance of appropriate choosing for the
constants of motion in the energy methods is pointed out. A set of
non-canonical constants of motion for the free rigid body on $\mathfrak{so}(5)$
and their role are analyzed.
