International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 707708

Let the basis B = (b_{1}, b_{2}, b_{3}) given by the scalar productsor by b_{1} = 2.449 (), b_{2} = b_{3} = 2.828 () (in arbitrary units), β_{23} = 60° (cos β_{23} = ½), β_{13} = 73.22° (), β_{12} = 64.34° ().
The aim is to find a standardized basis of shortest lattice vectors using Delaunay reduction. This example, given by B. Gruber (cf. Burzlaff & Zimmermann, 1985), shows the standardization problems remaining after the reduction.
The general reduction step can be described using Selling four flats. The corners are designated by the vectors a, b, c, d = −a − b − c. The edges are marked by the scalar products among these vectors. If positive scalar products can be found, choose the largest: (indicated as ab in Fig. 3.1.2.2a). The reduction transformation is: a_{D} = a, b_{D} = −b, c_{D} = c + b, d_{D} = d + b (see Fig. 3.1.2.2a). In this example, this results in the Selling four flat shown in Fig. 3.1.2.2(b). The next step, shown in Fig. 3.1.2.2(c), uses the (maximal) positive scalar product for further reduction. Finally, using b_{2} + b_{3} + b_{4} = −b_{1} we get the result shown in Fig. 3.1.2.2(d).

Delaunay reduction of Gruber's example (cf. Section 3.1.2.4). The edges of Selling tetrahedra are labelled by the scalar products of the vectors which designate the corners of the tetrahedra. 
The complete procedure can be expressed in a table, as shown in Table 3.1.2.4. Each pair of lines contains the starting basis and its scalar products before transformation as the first line, and then the transformed scalar products and the Delaunay basis after transformation below. In our case, four transformation steps are necessary. The result isThe final Selling tetrahedron shows that the Dirichlet domain belongs to Voronoi type 1. It fulfils no symmetry condition and thus corresponds to an anorthic (triclinic) lattice.

For further standardization we consider the Delaunay setAll bases of shortest lattice vectors () can be found:Any basis of shortest lattice vectors contains . For the vectors , , and are possible. can only be chosen from these vectors such that a linear independent triplet results.
The resulting five choices are given in Table 3.1.2.5. Any case corresponds to eight combinations of signs for the three basis vectors. The principle of the `homogenous corner' (i.e., there is always a pair of opposite corners of the corresponding cell where all angles are either nonacute or all three are acute) selects one of the bases in each case, thus five different bases remain. For the final choice the surfaces of the corresponding cells are given.

The maximal surface has cell No. 1 with the metrical parametersA last possibility for the standardization is the interchange of b and c with inversion of all basis vectors. In this way the sequence of β and γ can be interchanged:
References
Burzlaff, H. & Zimmermann, H. (1985). On the metrical properties of lattices. Z. Kristallogr. 170, 247–262.