  We examine how generalised geometries can be associated with a labelled
Dynkin diagram built around a gravity line. We present a series of new
generalised geometries based on the groups
$\mathit{Spin}(d,d)\times\mathbb{R}^+$ for which the generalised tangent space
transforms in a spinor representation of the group. In low dimensions these all
appear in subsectors of maximal supergravity theories. The case $d=8$ provides
a geometry for eight-dimensional backgrounds of M theory with only seven-form
flux, which have not been included in any previous geometric construction. This
geometry is also one of a series of "half-exceptional" geometries, which
"geometrise" a six-form gauge field. In the appendix, we consider examples of
other algebras appearing in gravitational theories and give a method to derive
the Dynkin labels for the "section condition" in general. We argue that
generalised geometry can describe restrictions and subsectors of many
gravitational theories.
