On the exceptional generalised Lie derivative for $d\ge 7$

In this work we revisit the $E_8\times {\mathbb{R}}^{+}$ generalised Lie derivative encoding the algebra of diffeomorphisms and gauge transformations of compactifications of M-theory on eight-dimensional manifolds, by extending certain features of the $E_7\times {\mathbb{R}}^{+}$ one. Compared to its $E_d\times {\mathbb{R}}^{+},d\le 7$ counterparts, a new term is needed for consistency. However, we find that no compensating parameters need to be introduced, but rather that the new term can be written in terms of the ordinary generalised gauge parameters by means of a connection. This implies that no further degrees of freedom, beyond those of the field content of the $E_8$ group, are needed to have a well defined theory. We discuss the implications of the structure of the $E_8\times {\mathbb{R}}^{+}$ generalised transformation on the construction of the $d=8$ generalised geometry. Finally, we suggest how to lift the generalised Lie derivative to eleven dimensions.