  It has been well established experimentally that the interplay of electronic
correlations and spin-orbit interactions in Ir$^{4+}$ and Ir$^{5+}$ oxides
results in insulating J$_{\rm eff}$=1/2 and J$_{\rm eff}$=0 ground states,
respectively. However, in compounds where the structural dimerization of iridum
ions is favourable, the direct Ir $d$--$d$ hybridisation can be significant and
takes a key role. Here, we investigate the effects of direct Ir $d$--$d$
hybridisation in comparison with electronic correlations and spin-orbit
coupling in Ba$_5$AlIr$_2$O$_{11}$, a compound with Ir dimers. Using a
combination of $ab$ $initio$ many-body wave function quantum chemistry
calculations and resonant inelastic X-ray scattering (RIXS) experiments, we
elucidate the electronic structure of Ba$_5$AlIr$_2$O$_{11}$. We find excellent
agreement between the calculated and the measured spin-orbit excitations.
Contrary to the expectations, the analysis of the many-body wave function shows
that the two Ir (Ir$^{4+}$ and Ir$^{5+}$) ions in the Ir$_2$O$_9$ dimer unit in
this compound preserve their local J$_{\rm eff}$ character close to 1/2 and 0,
respectively. The local point group symmetry at each of the Ir sites assumes an
important role, significantly limiting the direct $d$--$d$ hybridisation. Our
results emphasize that minute details in the local crystal field (CF)
environment can lead to dramatic differences in electronic states in iridates
and 5$d$ oxides in general.
