  Given an affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}[\sigma]$ of
arbitrary type, we determine certain minimal sets of annihilating fields of
standard $\tilde{\mathfrak{g}}[\sigma]$-modules. We then use these sets in
order to obtain a characterization of standard
$\tilde{\mathfrak{g}}[\sigma]$-modules in terms of irreducible loop
$\tilde{\mathfrak{g}}[\sigma]$-modules, which proves to be a useful tool for
combinatorial constructions of bases for standard
$\tilde{\mathfrak{g}}[\sigma]$-modules.
