  We use computer algebra to study polynomial identities for the trilinear
operation [a,b,c] = abc - acb - bac + bca + cab - cba in the free associative
algebra. It is known that [a,b,c] satisfies the alternating property in degree
3, no new identities in degree 5, a multilinear identity in degree 7 which
alternates in 6 arguments, and no new identities in degree 9. We use the
representation theory of the symmetric group to demonstrate the existence of
new identities in degree 11. The only irreducible representations of dimension
< 400 with new identities correspond to partitions 2^5 1 and 2^4 1^3 and have
dimensions 132 and 165. We construct an explicit new multilinear identity for
partition 2^5 1 and we demonstrate the existence of a new non-multilinear
identity in which the underlying variables are permutations of a^2 b^2 c^2 d^2
e^2 f.
