  Rotational bands are commonly used in the analysis of the spectra of atomic
nuclei. The early version of the interacting boson model of Arima and Iachello
has been foundational to the description of rotations in nuclei. The model is
based on a unitary spectrum generating algebra $U(6)$ and an orthogonal
(angular momentum) symmetry algebra $SO(3)$. A solvable limit of the model
contains $SU(3)$ in its dynamical symmetry chain. The corresponding Hamiltonian
is written as a linear combination of linear and quadratic Casimir invariants
of all the algebras in the chain. Prompted by these facts, a Hamiltonian
containing the $SU(3)$ quadratic Casimir squared is proposed to evaluate its
effects on rotational bands. The additional term yields three undetermined
parameters into the theory, which need be obtained from experiment. The lack of
data does not allow one to perform a detailed numerical analysis, but a rather
restricted one in terms of a one-parameter fit. Nevertheless, these additional
terms provide a good description of rotational bands of two nuclei of interest,
$^{156}\mathrm{Ga}$ and $^{234}\mathrm{U}$.
