  Rotational bands are commonplace in the spectra of atomic nuclei. Inspired by
early descriptions of these bands by quadrupole deformations of a liquid drop,
Elliott constructed a discrete nucleon representations of $\mathrm{SU}(3)$ from
fermionic creation and annihilation operators. Ever since, Elliott's model has
been foundational to descriptions of rotation in nuclei. Later work, however,
suggested the symplectic extension $\mathrm{Sp}(3,R)$ provides a more unified
picture. We decompose no-core shell-model nuclear wave functions into
symmetry-defined subspaces for several beryllium isotopes, as well as
$^{20}$Ne, using the quadratic Casimirs of both Elliott's $\mathrm{SU}(3)$ and
$\mathrm{Sp}(3,R)$. The band structure, delineated by strong $B(E2)$ values,
has a more consistent description in $\mathrm{Sp}(3,R)$ rather than
$\mathrm{SU}(3)$. {In particular, we confirm previous work finding in some
nuclides strongly connected upper and lower bands with the same underlying
symplectic structure.
