Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let $\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$ be a successive minima of $\Lambda$ with respect to the sup norm. Let $\lambda_{j} \leq U < \lambda_{j+1}$.

If we consider the set $S = \{ \bar{x} \in \Lambda : \| \bar{x} \| < U \}$, then by the definition of successive minima it follows that every point in $S$ must be a linear combination of $v_1, ..., v_j$. Since points achieving successive minima aren't necessarily a basis for the lattice, there can be points in $S$ where the coefficients in front of $v_i$'s are not integers. I am sure there is an example of a lattice $\Gamma$, where this happens, but I could not come up with one and I am interested in seeing an actual example. I would appreciate any assistance/comments! Thank you!

PS This question is related to my previous question, which was resolved Counting number of points in a lattice with bounded sup norm