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The branching rules between simple Lie algebras and its regular (maximal)simple subalgebras are studied. Two types of recursion relations for anomalousrelative multiplicities are obtained. One of them is proved to be thefactorized version of the other. The factorization property is based on theexistence of the set of weights $\Gamma$ specific for each injection. Thestructure of $\Gamma$ is easily deduced from the correspondence between theroot systems of algebra and subalgebra. The recursion relations thus obtainedgive rise to simple and effective algorithm for branching rules. The detailsare exposed by performing the explicit decomposition procedure for $A_{3}\oplus u(1) \to B_{4}$ injection.Comment: 15p.,LaTe

Recursion relations and branching rules for simple Lie algebras