the same results. Step 2 The next step is to compute the matrix elements of the operators and forming the basis of the Lie algebra of The matrix elements of and (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by where the labels and have been dropped since they are the same for all basis vectors in the representation. Due to the commutation relations the triple is a vector operator and the Wigner–Eckart theorem applies for computation of matrix elements between the states represented by the chosen basis. The matrix elements of where the superscript signifies that the defined quantities are the components of a spherical tensor operator of rank (which explains the factor as well) and the subscripts are referred to as in formulas below, are given by Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling with to get . The second factors are the reduced matrix elements. They do not depend on or , but depend on and, of course, . For a complete list of non-vanishing equations, see . Step 3 The next step is to demand that the Lie algebra relations hold, i.e. that This results in a set of equations for which the solutions are where Step 4 The imposition of the requirement of unitarity of the corresponding representation of the group restricts the possible values for the arbitrary complex numbers and . Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning This translates to leading to where is the angle of on polar form. For follows and is chosen by convention. There are two possible cases: In this case , real, This is the principal series. Its elements are denoted It follows: Since , is real and positive for , leading to . This is complementary series. Its elements are denoted This shows that the representations of above are all infinite-dimensional irreducible unitary representations. Explicit formulas Conventions and Lie algebra bases The metric of choice is given by , and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by: The commutation relations of the Lie algebra are: In three-dimensional notation,