The algebra AJ obtained from an associative algebra A on replacing the product xy
by x ⋅ y = 1∕2(xy + yx) is easily seen to be a Jordan algebra. Any subalgebra of a
Jordan algebra of this type is called special. It is known from work of Albert and
Paige that the kernel of the natural homomorphism from the free Jordan algebra on
three generators to the free special Jordan algebra on three generators is
nonzero and consequently that there exist three-variable relations which
hold identically in any homomorphic image of a special Jordan algebra but
which are not consequences of the defining identities (1). Such a relation we
shall call an S-identity. It is the purpose of this paper to establish that the
minimium possible degree for an S-identity is 8 and to give an example of an
S-identity of degree 8. In the final section we use an S-identity to give a short
proof of the main theorem of Albert and Paige in a slightly strengthened
form.
