Let \(M^{2j+2}\) be a closed Spin manifold, and let [, ] be the symmetric bilinear form on \(H^ j(M; {\mathbb{Z}}_ 2)\) defined by \([x,y]=\). In this paper, the authors study [, ] and the resulting cobordism invariants. For example, the rank modulo 2 of [, ] on integral cohomology is \(\) when dim M\(=8k+2\). The authors draw analogies with the work of \textit{G. Lusztig}, \textit{F. P. Peterson} and \textit{J. Milnor} [Topology 8, 357-359 (1969; Zbl 0165.263)]. The main result of the paper is: ``For a closed spin manifold \(M^{8k+2}\) and class \(z\in H^ 4(M; {\mathbb{Z}})\), the reduction mod 2 \(\rho\) z satisfies: \([\rho z,\rho z]=[\rho z,v_{4k}]\) where \(v_*\) is the Wu class of M.'' The proofs are based on computations of the spin cobordism groups of certain Eilenberg-MacLane spaces and related elaborate calculations.