  The Volume Operator plays a crucial role in the definition of the quantum
dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical
problems of LQG can therefore be performed only if one has sufficient control
over the volume spectrum. While closed formulas for the matrix elements are
currently available in the literature, these are complicated polynomials in 6j
symbols which in turn are given in terms of Racah's formula which is too
complicated in order to perform even numerical calculations for the
semiclassically important regime of large spins. Hence, so far not even
numerically the spectrum could be accessed. In this article we demonstrate that
by means of the Elliot -- Biedenharn identity one can get rid of all the 6j
symbols for any valence of the gauge invariant vertex, thus immensely reducing
the computational effort. We use the resulting compact formula to study
numerically the spectrum of the gauge invariant 4 -- vertex. The techniques
derived in this paper could be of use also for the analysis of spin -- spin
interaction Hamiltonians of many -- particle problems in atomic and nuclear
physics.
