  We define Jordan quadruple systems by the polynomial identities of degrees 4
and 7 satisfied by the Jordan tetrad {a,b,c,d} = abcd + dcba as a quadrilinear
operation on associative algebras. We find further identities in degree 10
which are not consequences of the defining identities. We introduce four
infinite families of finite dimensional Jordan quadruple systems, and construct
the universal associative envelope for a small system in each family. We obtain
analogous results for the anti-tetrad [a,b,c,d] = abcd - dcba. Our methods rely
on computer algebra, especially linear algebra on large matrices, the LLL
algorithm for lattice basis reduction, representation theory of the symmetric
group, noncommutative Grobner bases, and Wedderburn decompositions of
associative algebras.
