On the different formulations of the E11 equations of motion

The nonlinear realization of the semi-direct product of Ell with its vector representation leads to equation of motions for the field graviton, three-form, six-form, dual graviton and the level four fields which correctly describe the degree of freedom of 11-dimensional supergravity at the linearized level. The equations with one derivative generically hold as equivalence relations and are often duality relations. From these equations, by taking derivatives, one can derive equations that are equations of motion of the familiar kind. The entire hierarchy of equations is Ell invariant and the construction does not require any steps beyond E-11. We review these past developments with an emphasis on the features that have been overlooked in hep-th:1703.01305, whose alternative approach we also comment on.