  We briefly review why the non-linear realisation of the semi-direct product
of a group with one of its representations leads to a field theory defined on a
generalised space-time equipped with a generalised vielbein. We give formulae,
which only involve matrix multiplication, for the generalised vielbein, the
Cartan forms and their transformations. We consider the generalised space-time
introduced in 2003 in the context of the non-linear realisation of the
semi-direct product of E(11) and its first fundamental representation. For this
latter theory we give explicit expressions for the generalised vielbein up to
and including the levels associated with the dual graviton in four, five and
eleven dimensions and for the IIB theory in ten dimensions. We also compute the
generalised vielbein, up to the analogous level, for the non-linear realisation
of the semi-direct product of very extended SL(2) with its first fundamental
representation, which is a theory associated with gravity in four dimensions.
