  The electronic structure of Sr$_3$CuIrO$_6$, a model system for the 5d Ir ion
in an octahedral environment, is studied through a combination of resonant
inelastic x-ray scattering (RIXS) and theoretical calculations. RIXS spectra at
the Ir L$_3$-edge reveal an Ir $t_{2g}$ manifold that is split into three
levels, in contrast to the expectations of the strong spin-orbit-coupling
limit. Effective Hamiltonian and $ab inito$ quantum chemistry calculations find
a strikingly large non-cubic crystal field splitting comparable to the
spin-orbit coupling, which results in a strong mixing of the
$j_{\mathsf{eff}}=1/2$ and $j_{\mathsf{eff}}=3/2$ states and modifies the
isotropic wavefunctions on which many theoretical models are based.
