International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 89

A general monoclinic transformation is of the formwith a diagonal matrix whose entries are or , and a vector whose entries are 0 or . We may thus decompose both real and reciprocal space into a direct sum of a subspace where acts as the identity, and a subspace where acts as minus the identity, with . All usual entities may be correspondingly written as direct sums, for instance:
We will use factoring by 2, with decimation in frequency when computing structure factors, and decimation in time when computing electron densities; this corresponds to with , . The nonprimitive translation vector then belongs to , and thusThe symmetry relations obeyed by and F are as follows: for electron densitiesor, after factoring by 2,while for structure factorswith its Friedel counterpartor, after factoring by 2,with Friedel counterpart
When calculating electron densities, two methods may be used.
References
Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.