Referring an ADP tensor to a Cartesian basis is somewhat less simple. We proceed to show how this is done, both in order to illustrate the above-outlined transformation of tensors and to provide some background for the following section.
Construct a Cartesian system by taking, for example, 
along a, 
along 
, and 
along
the vector product 
. The resulting
vectors
comprise an appropriate and common orthonormal set of basis vectors.
Eqs. (2.1.35) and (2.1.17) are both expressions for u. Hence
If we take the scalar products of the left-hand side and right-hand sides of
eq. (2.1.40) with , and , we
obtain three linear equations, or a matrix equation of the form
This is a transformation of the components of u, referred to the basis of the direct lattice, to its Cartesian components. The transformation matrix can be evaluated once the Cartesian basis vectors are defined (e.g., as above). If we adopt the index notation in the second line of eq. (2.1.17), eq. (2.1.41) can be written as
with
We can similarly transform a product of components of u between the lattice and Cartesian bases, and finally an average of such a product:
If we now make use of eqs. (2.1.22) and (2.1.37) we obtain
which is the expression for the transformation of
to Cartesian coordinates.
The orthogonalization of U proceeds along similar lines. The required version of eq. (2.1.40) is now [cf. eq. (2.1.23)]
and following the same procedure by which eq. (2.1.41) is obtained, we arrive at the transformation
which relates the components of u, referred to the 
basis, to its Cartesian components. Equation (2.1.46)
can be written concisely as
with
an element of the matrix product appearing in eq. (2.1.46). The desired transformation is obtained analogously to eq. (2.1.44) as
with 
and
.
The explicit form of the transformation matrix appearing in eq. (2.1.41), for the specific Cartesian basis defined in eqs. (2.1.39), is:
Of course, a Cartesian basis associated with the direct and/or reciprocal bases can be chosen in an unlimited number of ways. A more general discussion of the construction of such Cartesian bases is given elsewhere (Shmueli, 1993).
