  Let $\mathit{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$, where $d>1$ is a cube-free
positive integer, $\mathit{k}_0=\mathbb{Q}(\zeta_3)$ be the cyclotomic field
containing a primitive cube root of unity $\zeta_3$, and
$G=\operatorname{Gal}(\mathit{k}/\mathit{k}_0)$. The possible prime
factorizations of $d$ in our main result [2, Thm. 1.1] give rise to new
phenomena concerning the chain $\Theta=(\theta_i)_{i\in\mathbb{Z}}$ of
\textit{lattice minima} in the underlying pure cubic subfield
$L=\mathbb{Q}(\sqrt[3]{d})$ of $\mathit{k}$. The aims of the present work are
to give criteria for the occurrence of generators of primitive ambiguous
principal ideals
$(\alpha)\in\mathcal{P}_{\mathit{k}}^G/\mathcal{P}_{\mathit{k}_0}$ among the
lattice minima $\Theta=(\theta_i)_{i\in\mathbb{Z}}$ of the underlying pure
cubic field $L=\mathbb{Q}(\sqrt[3]{d})$, and to explain exceptional behavior of
the chain $\Theta$ for certain radicands $d$ with impact on determining the
principal factorization type of $L$ and $\mathit{k}$ by means of Voronoi's
algorithm.
