  The dual Steenrod algebra has a canonical subalgebra isomorphic to the
homology of the Brown-Peterson spectrum. We will construct a secondary
operation in mod-2 homology and show that this canonical subalgebra is not
closed under it. This allows us to conclude that the 2-primary Brown-Peterson
spectrum does not admit the structure of an E_n-algebra for any n greater than
or equal to 12, answering a question of May in the negative.
