  In this article, we prove the theorems concerning the trace relation of
SO(3), SU(2), and SO(n) which are representation of SO(3) and SU(2). An
interesting fact we found is the trace relation of SU(2) gives the spherical
law of cosine which in turns is a dihedral angle relation, a constraint that
must be satisfied by closed Euclidean simplices. Moreover, we applied our
results on general group elements to holonomies on the simplicial complex of
Regge Calculus, which is the main motivation of this article. Here, we found
that: (1) in 4-dimensional Euclidean Regge Gravity, all the holonomy circling a
single hinge are simple rotations, and (2) the dihedral angle relation
represents the 'contracted' Bianchi identity for a simplicial complex.
