  We present atomistic computations within an empirical pseudopotential
framework for the electron $s$-shell ground state $g$ tensor of InGaAs quantum
dots (QDs) embedded to host matrices that grant electronic confinement. A large
structural set consisting of geometry, size, and molar fraction variations is
worked out which also includes a few representative uniform strain cases. The
tensor components are observed to display insignificant discrepancies even for
the highly anisotropic shapes. The family of $g$-factor curves associated with
these parameter combinations coalesces to a single universal one when plotted
as a function of the gap energy, thus confirming a recent assertion reached
under much restrictive conditions. Our work extends its validity to alloy QDs
with various shapes and finite confinement that allows for penetration to the
host matrix, placing it on a more realistic basis. Accordingly, the electrons
in InGaAs QDs having $s$-shell transition energies close to 1.13 eV will be
least susceptible to magnetic field. We also show that low indium concentration
offer limited $g$-factor tunability under shape or confinement variations.
These findings can be taken into consideration in the fabrication and the use
of InGaAs QDs with $g$-near-zero or other targeted $g$ values for spintronic or
electron spin resonance-based direct quantum logic applications.
