  Given an arbitrary basis for a mathematical lattice, to find a ``good" basis
for it is one of the classic and important algorithmic problems. In this note,
we give a new and simpler proof of a theorem by Regavim (arXiv:2106.03183): we
construct a 18-dimensional lattice that does not have a basis that satisfies
the following two properties simultaneously: 1. The basis includes the shortest
non-zero lattice vector. 2. The basis is shortest, that is, minimizes the
longest basis vector (alternatively: the sum or the sum-of-squares of the basis
vectors). The vectors' length can be measured in any $\ell^q$ norm, for $q\in
\mathbb{N}_+$ (albeit, via another lattice, of a somewhat larger dimension).
