  We investigate the superalgebra of derivations generated by the fundamental
forms on manifolds with reduced structure group. In particular, we point out a
relation between the algebra of derivations of heterotic geometries that admit
Killing spinors and the commutator algebra of holonomy symmetries in sigma
models. We use this to propose a Lie bracket on the space of fundamental forms
of all heterotic geometries with a non-compact holonomy group and present the
associated derivation algebras. We also explore the extension of these results
to heterotic geometries with compact holonomy groups and, more generally, to
manifolds with reduced structure group. A brief review of the classification of
heterotic geometries that admit Killing spinors and an extension of this
classification to some heterotic inspired geometries are also included.
