  Recently Ohsawa has studied the Marsden-Weinstein-Meyer quotient of the
manifold $T^*\mathbb{R}^n\times T^*\mathbb{R}^{2n^2}$ under a
$\operatorname{O}(2n)$-symmetry, and has used this quotient to describe the
relationship between two different parametrisations of Gaussian wave packet
dynamics commonly used in semiclassical mechanics. In this paper we suggest a
new interpretation of (a subset of) the unreduced space as being the frame
bundle $\mathcal{F}(T^*\mathbb{R}^n)$ of $T^*\mathbb{R}^n$. We outline some
advantages of this interpretation, and explain how it can be extended to more
general symplectic manifolds using the notion of the diagonal lift of a
symplectic form due to Cordero and de Le\'on.
