  We determine the multiplicity of the irreducible representation V(n) of the
simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power
$\Lambda^4 V(n)$. 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 $\le 7$ satisfied by the sl(2,C)-invariant alternating
quaternary algebra structures obtained from the projections $\Lambda^4 V(n) \to
V(n)$. 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.
