Alternating quaternary algebra structures on irreducible representations of sl2(C)

We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl2(C) as a direct summand of its fourth exterior power Λ4V(n). The multiplicity is 1 (resp. 2) if and only if n=4,6 (resp. n=8,10). For these n we determine the multilinear polynomial identities of degree ⩽7 satisfied by the sl2(C)-invariant alternating quaternary algebra structures obtained from the projections Λ4V(n)→V(n). We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.