Linear idempotents in Matsuo algebras

Matsuo algebras are an algebraic incarnation of 3-transposition groups with a parameter $\alpha$, where idempotents takes the role of the transpositions. We show that a large class of idempotents in Matsuo algebras satisfy the Seress property, making these nonassociative algebras well-behaved analogously to associative algebras, Jordan algebras and vertex (operator) algebras. We calculate eigenvalues in the Matsuo algebra of ${\rm Sym}(n)$ for any $\alpha$, generalising some vertex algebra results for which $\alpha=\frac{1}{4}$. Finally, in the Matsuo algebra of the root system ${\rm D}_n$, we show $n-3$ conjugacy classes of involutions coming from the Weyl group are in natural bijection with idempotents in the algebra via their fusion rules.