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We continue our exploration of the extent to which the spectrum encodes thelocal geometry of a locally homogeneous three-manifold and find that if $(M,g)$and $(N,h)$ are a pair of locally homogeneous, locally non-isometricisospectral three-manifolds, where $M$ is an elliptic three-manifold, then$(1)$ $N$ is also an elliptic three-manifold, $(2)$ $M$ and $N$ havefundamental groups of different orders, $(3)$ $(M,g)$ and $(N,h)$ both havenon-degenerate Ricci tensors and $(4)$ the metrics $g$ and $h$ are sufficientlyfar from a metric of constant sectional curvature. We are unaware of any suchisospectral pair and such a pair could not arise via the classical Sunadamethod. As part of the proof, we provide an explicit description of theisometry group of a compact simple Lie group equipped with a left-invariantmetric---improving upon the results of Ochiai-Takahashi and Onishchik---whichwe use to classify the locally homogeneous metrics on an ellipticthree-manifold $\Gamma \backslash S^3$ and we determine that any collection ofisospectral locally homogeneous metrics on an elliptic three-manifold consistsof at most two isometry classes that are necessarily locally isometric. Inparticular, the left-invariant metrics on $\operatorname{SO}(3)$ (respectively,$S^3$) can be mutually distinguished via their spectra. The previous statementhas the following interpretation in terms of physical chemistry: the moments ofinertia of a molecule can be recovered from its rotational spectrum.

Geometric structures and the Laplace spectrum, part II