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Special identities for quasi-Jordan algebras

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities $a(bc) = a(cb)$, $(ba)a^2 = (ba^2)a$, and $(b,a^2,c) = 2(b,a,c)a$. These identities are satisfied by the product $ab = a \dashv b + b \vdash a$ in an associative dialgebra. We use computer algebra to show that every identity for this product in degree $\le 7$ is a consequence of the three identities in degree $\le 4$, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.