real). The Clebsch–Gordan coefficients can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to the requirement that the norm of the state must be one. The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with . Repeated use of that equation gives all coefficients. This procedure to find the Clebsch–Gordan coefficients shows that they are all real in the Condon–Shortley phase convention. Explicit expression Orthogonality relations These are most clearly written down by introducing the alternative notation The first orthogonality relation is (derived from the fact that ) and the second one is Special cases For the Clebsch–Gordan coefficients are given by For and we have For and we have For we have For , we have For we have Symmetry properties A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using . The symmetry properties of Wigner 3-j symbols are much simpler. Rules for phase factors Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore is not necessarily for a given quantum number unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule: for any angular-momentum-like quantum number . Nonetheless, a combination of and is always an integer, so the stronger rule applies for these combinations: This identity also holds if the sign of either or or both is reversed. It is useful to observe that any phase factor for a given pair can be reduced to the canonical form: where and (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this form is only locally canonical: it fails to take into account the rules that govern combinations of pairs such as the one described in the next paragraph.) An additional rule holds for combinations of , , and that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol: This identity also holds if the sign of any is reversed, or if any of them are substituted with an instead. Relation to Wigner 3-j symbols Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations. The factor is due to the Condon–Shortley