  We present the first detailed elemental abundances in the ultra-faint
Magellanic satellite galaxies Carina II (Car II) and Carina III (Car III). With
high-resolution Magellan/MIKE spectroscopy, we determined abundances of nine
stars in Car II including the first abundances of an RR Lyrae star in an
ultra-faint dwarf galaxy; and two stars in Car III. The chemical abundances
demonstrate that both systems are clearly galaxies and not globular clusters.
The stars in these galaxies mostly display abundance trends matching those of
other similarly faint dwarf galaxies: enhanced but declining [alpha/Fe] ratios,
iron-peak elements matching the stellar halo, and unusually low neutron-capture
element abundances. One star displays a low outlying [Sc/Fe] = -1.0. We detect
a large Ba scatter in Car II, likely due to inhomogeneous enrichment by
low-mass AGB star winds. The most striking abundance trend is for [Mg/Ca] in
Car II, which decreases from +0.4 to -0.4 and indicates clear variation in the
initial progenitor masses of enriching core-collapse supernovae. So far, the
only ultra-faint dwarf galaxies displaying a similar [Mg/Ca] trend are likely
satellites of the Large Magellanic Cloud. We find two stars with [Fe/H] < -3.5,
whose abundances likely trace the first generation of metal-free Population III
stars and are well-fit by Population III core-collapse supernova yields. An
appendix describes our new abundance uncertainty analysis that propagates
line-by-line stellar parameter uncertainties.
