  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.
