  We present a spectral analysis of the binary G 224-58 AB that consists of the
coolest M extreme subdwarf (esdM5.5) and a brighter primary (esdK5). This
binary may serve as a benchmark for metallicity measurement calibrations and as
a test-bed for atmospheric and evolutionary models for esdM objects.
  We determine abundances primarily using high resolution optical spectra of
the primary. Other parameters were determined from the fits of synthetic
spectra computed with these abundances to the observed spectra from 0.4 to 2.5
microns for both components.
  We determine \Tef =4625 $\pm$ 100 K, \logg = 4.5 $\pm$ 0.5 for the A
component and \Tef = 3200 $\pm$ 100 K, \logg = 5.0 $\pm$ 0.5, for the B
component. We obtained abundances of [Mg/H]=$-$1.51$\pm$0.08,
[Ca/H]=$-$1.39$\pm$0.03, [Ti/H]=$-$1.37$\pm$0.03 for alpha group elements and
[CrH]=$-$1.88$\pm$0.07, [Mn/H]=$-$1.96$\pm$0.06, [Fe/H]=$-$1.92$\pm$0.02,
[Ni/H]=$-$1.81$\pm$0.05 and [Ba/H]W=$-$1.87$\pm$0.11 for iron group elements
from fits to the spectral lines observed in the optical and infrared spectral
regions of the primary. We find consistent abundances with fits to the
secondary albeit at lower signal-to-noise.
  Abundances of elements in \ga and \gb atmospheres cannot be described by one
metallicity parameter. The offset of $\sim$ 0.4 dex between the abundances
derived from alpha element and iron group elements corresponds with our
expectation for metal-deficient stars. We thus clarify that some indices used
to date to measure metallicities for establishing esdM stars based on CaH, MgH
and TiO band system strength ratios in the optical and H$_2$O in the infrared
relate to abundances of alpha-element group rather than to iron peak elements.
For metal deficient M dwarfs with [Fe/H] < -1.0, this provides a ready
explanation for apparently inconsistent "metallicities" derived using different
methods.
