International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.5, p. 84

The metric tensor of a crystal lattice with a basis is the (3 × 3) matrixwhich can formally be described as(cf. Section 1.3.2 ). The transformation of the metric tensor under the coordinate transformation follows directly from its definition:where is the transposed matrix of P. The transformation behaviour of G under is determined by the matrix P, i.e. G is not affected by an origin shift p.
The volume V of the unit cell defined by the basis vectors can be obtained from the determinant of the metric tensor, . The transformation behaviour of V under a coordinate transformation follows from the transformation behaviour of the metric tensor [note that ]: = = = = , i.e.which is reduced to if .
Similarly, the metric tensor of the reciprocal lattice and the volume of the unit cell defined by the basis vectors transform asAgain, it is only the linear part that determines the transformation behaviour of and under coordinate transformations.