An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that

1. $\{b_1,…,b_n \}$ is a basis,
2. $b_1+\cdots b_{n+1}=0$,
3. $q_{ij}=b_i^Tb_j\le 0$ for all $i\ne j$. The $q_{ij}$ are called selling parameters.

Condition (2) is called the superbase condition and condition (3) the obtuse condition.

Given a lattice which is known to be Voronoi's first kind, are any methods known to find a superbase for it?

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that

1. $\{b_1,\ldots,b_n \}$ is a basis,
2. $b_1+\cdots +b_{n+1}=0$,
3. $q_{ij}=b_i^Tb_j\le 0$ for all $i\ne j$. The $q_{ij}$ are called selling parameters.

Condition (2) is called the superbase condition and condition (3) the obtuse condition.

Given a lattice which is known to be Voronoi's first kind, are any methods known to find a superbase for it?