  We examine the structure of gauge transformations in extended geometry, the
framework unifying double geometry, exceptional geometry, etc. This is done by
giving the variations of the ghosts in a Batalin-Vilkovisky framework, or
equivalently, an $L_\infty$ algebra. The $L_\infty$ brackets are given as
derived brackets constructed using an underlying Borcherds superalgebra ${\scr
B}({\mathfrak g}_{r+1})$, which is a double extension of the structure algebra
${\mathfrak g}_r$. The construction includes a set of "ancillary" ghosts. All
brackets involving the infinite sequence of ghosts are given explicitly. All
even brackets above the 2-brackets vanish, and the coefficients appearing in
the brackets are given by Bernoulli numbers. The results are valid in the
absence of ancillary transformations at ghost number 1. We present evidence
that in order to go further, the underlying algebra should be the corresponding
tensor hierarchy algebra.
