Matsuo algebras in characteristic 2

We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. Instead of idempotent elements associated to points in the corresponding Fischer space, our algebras are now generated by nilpotent elements associated to lines. For many 3-transposition groups, this still gives rise to a $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group $\mathrm{Sym}(4)$, the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group ${\mathbb{G}}_a^2⋊{\mathbb{G}}_m$.