H-spaces analogous to E8mod 3

Let p be an odd prime. Let X0 be a finite, p-local, simply connected homotopy associative H-space. Suppose H*(X0;ℤp) contains the subalgebra [inline-graphic not available: see fulltext] satisfying [inline-graphic not available: see fulltext] The only known examples occur for p=3 and involve the Lie group E8. In this note we prove that if X0 exists, then p must be 3.