  The recent investigation of the gauge structure of extended geometry is
generalised to situations when ancillary transformations appear in the
commutator of two generalised diffeomorphisms. The relevant underlying
algebraic structure turns out to be a tensor hierarchy algebra rather than a
Borcherds superalgebra. This tensor hierarchy algebra is a non-contragredient
superalgebra, generically infinite-dimensional, which is a double extension of
the structure algebra of the extended geometry. We use it to perform a
(partial) analysis of the gauge structure in terms of an $L_\infty$ algebra for
extended geometries based on finite-dimensional structure groups. An invariant
pseudo-action is also given in these cases. We comment on the continuation to
infinite-dimensional structure groups. An accompanying paper deals with the
mathematical construction of the tensor hierarchy algebras.
