  We prove that if the associated fourth order tensor of a quadratic form has a
linear elastic cubic symmetry then it is quasiconvex if and only if it is
polyconvex, i.e. a sum of convex and null-Lagrangian quadratic forms. We prove
that allowing for slightly less symmetry, namely only cyclic and
axis-reflection symmetry, gives rise to a class of extremal quasiconvex
quadratic forms, that are not polyconvex. Non-affine boundary conditions on the
potential are identified which allow one to obtain sharp bounds on the
integrals of these extremal quasiconvex quadratic forms of $\nabla u$ over an
arbitrary region.
