  We employ first principles total energy and phonon calculations to address
the structure and stability of Al$_2$Fe. This structure, which is reported as
stable in the assessed Al-Fe phase diagram, is distinguished by an unusually
low triclinic symmetry. The initial crystallographic structure determination
additionally featured an unusual hole large enough to accommodate an additional
atom. Our calculations indicate the hole must be filled, but predict the
triclinic structure is unstable relative to a simpler structure based on the
prototype MoSi$_2$. This MoSi$_2$ structure is interesting because it is
predicted to be nonmagnetic, electrically insulating and high density, while
the triclinic structure is magnetic, metallic and low density. We reconcile
this seeming contradiction by demonstrating a high vibrational entropy that
explains why the triclinic structure is stable at high temperatures. Finally,
we note that Al$_5$Fe$_2$ poses a similar problem of unexplained stability.
