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In this article, we prove the theorems concerning the trace relation ofSO(3), SU(2), and SO(n) which are representation of SO(3) and SU(2). Aninteresting fact we found is the trace relation of SU(2) gives the sphericallaw of cosine which in turns is a dihedral angle relation, a constraint thatmust be satisfied by closed Euclidean simplices. Moreover, we applied ourresults on general group elements to holonomies on the simplicial complex ofRegge Calculus, which is the main motivation of this article. Here, we foundthat: (1) in 4-dimensional Euclidean Regge Gravity, all the holonomy circling asingle hinge are simple rotations, and (2) the dihedral angle relationrepresents the 'contracted' Bianchi identity for a simplicial complex.

Contracted Bianchi Identity and Angle Relation on n-dimensional Simplicial Complex of Regge Calculus