  Sr$_{2}$IrO$_{4}$ has often been described via a simple, one-band pseudo-spin
1/2 model, subject to electron-electron interactions, on a square lattice,
fostering analogies with cuprate superconductors, believed to be well described
by a similar model. In this work we argue - based on a detailed study of the
low-energy electronic structure by circularly polarized spin and angle-resolved
photoemission spectroscopy combined with dynamical mean-field theory
calculations - that a pseudo-spin 1/2 model fails to capture the full
complexity of the system. We show instead that a realistic multi-band Hubbard
Hamiltonian, accounting for the full correlated $t_{2g}$ manifold, provides a
detailed description of the interplay between spin-orbital entanglement and
electron-electron interactions, and yields quantitative agreement with
experiments. Our analysis establishes that the $j_{3/2}$ states make up a
substantial percentage of the low energy spectral weight, i.e. approximately
74% as determined from the integration of the $j$-resolved spectral function in
the $0$ to $-1.64$ eV energy range. The results in our work are not only of
relevance to iridium based materials, but more generally to the study of
multi-orbital materials with closely spaced energy scales.
