  Metals can be defined as materials with a Fermi surface or as materials
exhibiting metallic conduction (i.e., $\mathrm{d} \rho / \mathrm{d}T > 0$).
Usually, these definitions both hold at low temperatures, such as liquid-helium
temperatures, as the Fermi energy is sufficiently larger than the thermal
energy. However, they may not both hold in correlated electron systems where
the Fermi energy is reduced by renormalization. In this paper, we demonstrate
that although the resistivity of CeFe$_2$Al$_{10}$ increases with decreasing
temperature below $\sim20$ K, CeFe$_2$Al$_{10}$ is a metal with a Fermi
surface. This assertion is based on the observation of Shubnikov--de Haas
oscillations and a Hall resistivity that changes sign with the magnetic field,
which requires the coexistence of electron and hole carriers. Our analysis of
Shubnikov--de Haas and magnetotransport data indicates that the Fermi energies
are as small as $\sim$30 K and that, despite the increasing carrier mobility
with decreasing temperature as in conventional metals, the loss of thermally
excited carriers leads to nonmetallic conduction ($\mathrm{d} \rho /
\mathrm{d}T < 0$) below $\sim20$ K. Furthermore, we investigate how this
anomalous metal transforms to a more conventional metal with metallic
conduction by the application of high pressure and a high magnetic field. This
study illustrates the subtle distinction between semimetals and semiconductors
in correlated electron systems. This distinction is relevant to investigations
of correlated topological insulators and semimetals.
