Generalized Racah coefficient, designated as U coefficient in this paper, has been defined as the transformation function between two different coupling schemes in pairs of any four angular momenta, corresponding to the Racah coefficient defined as the transformation function between two different coupling scheme of any three angular momenta. Several simple properties of the U coefficient have been derived, and the method of tensor operators made to be extended to more general problems. Transformation coefficients between LS- and jj-coupling schemes in a many particle system can be evaluated by making use of these coefficients.