  We give a description of the $\mathrm{GL}_4$ Rapoport-Zink space, including
the connected components, irreducible components, intersection behavior of the
irreducible components, and Ekedahl-Oort stratification. As an application of
this, we also give a description of the supersingular locus of the Shimura
variety for the group $\mathrm{GU}(2,2)$ over a prime split in the relevant
imaginary quadratic field.
