  In this paper, we aim to derive high-precision alpha-element abundances using
CRIRES high-resolution IR spectra of 72 cool M giants of the inner Galactic
bulge. Silicon, magnesium, and calcium abundances were determined by fitting a
synthetic spectrum for each star. We also incorporated recent theoretical data
into our spectroscopic analysis (i.e. updated K-band line list, better
broadening parameter estimation, non-local thermodynamic equilibrium (NLTE)
corrections). We compare these inner bulge alpha abundance trends with those of
solar neighbourhood stars observed with IGRINS using the same line list and
analysis technique; we also compare our sample to APOGEE DR17 abundances for
inner bulge stars. We investigate bulge membership using spectro-photometric
distances and orbital simulations. We construct a chemical-evolution model that
fits our metallicity distribution function (MDF) and our alpha-element trends.
Among our 72 stars, we find four that are not bulge members. [Si/Fe] and
[Mg/Fe] versus [Fe/H] trends show a typical thick disc alpha-element behaviour,
except that we do not see any plateau at supersolar metallicities as seen in
other works. The NLTE analysis lowers [Mg/Fe] typically by $\sim$0.1 dex,
resulting in a noticeably lower trend of [Mg/Fe] versus [Fe/H]. The derived
[Ca/Fe] versus [Fe/H] trend has a larger scatter than those for Si and Mg, but
is in excellent agreement with local thin and thick disc trends. With our
updated analysis, we constructed one of the most detailed studies of the alpha
abundance trends of cool M giants in the inner Galactic bulge. We modelled
these abundances by adopting a two-infall chemical-evolution model with two
distinct gas-infall episodes with timescales of 0.4 Gyr and 2 Gyr,
respectively. Based on a very meticulous spectral analysis, we have constructed
detailed and precise chemical abundances of Mg, Si, and Ca for cool M giants.
