  We show that a symplectic reduction of the symmetric representation of the
generalized $n$-dimensional rigid body equations yields the $n$-dimensional
Euler equation. This result provides an alternative to the more elaborate
relationship between these equations established by Bloch, Crouch, Marsden, and
Ratiu. Specifically, we exploit the inherent
$\mathsf{Sp}(2n,\mathbb{R})$-symmetry in the symmetric representation to
present its relationship with the Euler equation via symplectic reduction
facilitated by the dual pair recently developed by Skerritt and Vizman.
