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

The Symmetric Representation of the Generalized Rigid Body Equations and Symplectic Reduction