  We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) =
\mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular
locus in characteristic two. We show that the locus of supersingular Jacobians
$\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimension
three. One way to obtain that result uses an analysis of the data of smooth
genus four curves and principally polarized abelian fourfolds defined over
$\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using
the indecomposability of some $\mathrm{BT}_1$ group schemes.
