  Iridates with the 5$d^4$ electronic configuration have attracted recent
interest due to reports of magnetically-ordered ground states despite
longstanding expectations that their strong spin-orbit coupling would generate
a $J = 0$ electronic ground state for each Ir$^{5+}$ ion. The major focus of
prior research has been on the double perovskite iridates Ba$_2$YIrO$_6$ and
Sr$_2$YIrO$_6$, where the nature of the ground states (i.e. ordered vs
non-magnetic) is still controversial. Here we present neutron powder
diffraction, high energy resolution fluorescence detected x-ray absorption
spectroscopy (HERFD-XAS), resonant inelastic x-ray scattering (RIXS), magnetic
susceptibility, and muon spin relaxation data on the related double perovskite
iridates Ba$_2$LuIrO$_6$, Sr$_2$LuIrO$_6$, Ba$_2$ScIrO$_6$, and Sr$_2$ScIrO$_6$
that enable us to gain a general understanding of the electronic and magnetic
properties for this family of materials. Our HERFD-XAS and RIXS measurements
establish $J = 0$ electronic ground states for the Ir$^{5+}$ ions in all cases,
with similar values for Hund's coupling $J_{\rm H}$ and the spin-orbit coupling
constant $\lambda_{\rm SOC}$. Our bulk susceptibility and muon spin relaxation
data find no evidence for long-range magnetic order or spin freezing, but they
do reveal weak magnetic signals that are consistent with extrinsic local
moments. Our results indicate that the large $\lambda_{\rm SOC}$ is the key
driving force behind the electronic and magnetic ground states realized in the
5$d^4$ double perovskite iridates, which agrees well with conventional wisdom.
