We compute the cotorsion product of the mod 2 cohomology of spinor
group spin(n), which is the E2–term of the
Rothenberg–Steenrod spectral sequence for the mod 2 cohomology of
the classifying space of the spinor group spin(n). As a
consequence of this computation, we show the non-collapsing of the
Rothenberg–Steenrod spectral sequence for n≥ 17.