International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 3839
Section 2.2.16.2. Settings^{a}Institut für Kristallographie, RheinischWestfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands 
The term setting of a cell or of a space group refers to the assignment of labels (a, b, c) and directions to the edges of a given unit cell, resulting in a set of basis vectors a, b, c. (For orthorhombic space groups, the six settings are described and illustrated in Section 2.2.6.4.)
The symbol for each setting is a shorthand notation for the transformation of a given starting set abc into the setting considered. It is called here `setting symbol'. For instance, the setting symbol bca stands for or where a′, b′, c′ is the new set of basis vectors. (Note that the setting symbol bca does not mean that the old vector a changes its label to b, the old vector b changes to c, and the old c changes to a.) Transformation of one setting into another preserves the shape of the cell and its orientation relative to the lattice. The matrices of these transformations have one entry 1 or −1 in each row and column; all other entries are 0.
In monoclinic space groups, one axis, the monoclinic symmetry direction, is unique. Its label must be chosen first and, depending upon this choice, one speaks of `unique axis b', `unique axis c' or `unique axis a'.^{12} Conventionally, the positive directions of the two further (`oblique') axes are oriented so as to make the monoclinic angle nonacute, i.e. , and the coordinate system righthanded. For the three cell choices, settings obeying this condition and having the same label and direction of the unique axis are considered as one setting; this is illustrated in Fig. 2.2.6.4.
Note: These three cases of labelling the monoclinic axis are often called somewhat loosely baxis, caxis and aaxis `settings'. It must be realized, however, that the choice of the `unique axis' alone does not define a single setting but only a pair, as for each cell the labels of the two oblique axes can be interchanged.
Table 2.2.16.1 lists the setting symbols for the six monoclinic settings in three equivalent forms, starting with the symbols a b c (first line), a b c (second line) and a b c (third line); the unique axis is underlined. These symbols are also found in the headline of the synoptic Table 4.3.2.1 , which lists the spacegroup symbols for all monoclinic settings and cell choices. Again, the corresponding transformation matrices are listed in Table 5.1.3.1 .

In the spacegroup tables, only the settings with b and c unique are treated and for these only the lefthand members of the double entries in Table 2.2.16.1. This implies, for instance, that the caxis setting is obtained from the baxis setting by cyclic permutation of the labels, i.e. by the transformation In the present discussion, also the setting with a unique is included, as this setting occurs in the subgroup entries of Part 7 and in Table 4.3.2.1 . The aaxis setting is obtained from the caxis setting also by cyclic permutation of the labels and from the baxis setting by the reverse cyclic permutation: .
By the conventions described above, the setting of each of the cell choices 1, 2 and 3 is determined once the label and the direction of the uniqueaxis vector have been selected. Six of the nine resulting possibilities are illustrated in Fig. 2.2.6.4.