Transchromatic extensions in motivic bordism

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGL and of the Real bordism spectrum MU_{\mathbb{R}}. These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height (n+1). Using these, we deduce a family of exotic multiplications in the \pi _{(\ast ,\ast )}MGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the \pi _{\star }MU_\mathbb{R}-module structure on the Real Morava K-theories.