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 $\mathbb{ Z}/2\mathbb{ Z}$-graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group $\operatorname{ Sym}(4)$, the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group $\mathbb{ G}_a^2 \rtimes \mathbb{ G}_m$.
Categories: math.GR math.RA