  Synchrotron X-ray diffraction experiment shows that the metal-insulator
transition occurring in a ferromagnetic state of a hollandite
K$_2$Cr$_8$O$_{16}$ is accompanied by a structural distortion from the
tetragonal $I4/m$ to monoclinic $P112_{1}/a$ phase with a
$\sqrt{2}\times\sqrt{2}\times 1$ supercell. Detailed electronic structure
calculations demonstrate that the metal-insulator transition is caused by a
Peierls instability in the quasi-one-dimensional column structure made of four
coupled Cr-O chains running in the $c$-direction, leading to the formation of
tetramers of Cr ions below the transition temperature. This furnishes a rare
example of the Peierls transition of fully spin-polarized electron systems.
