  In a recent study, we have observed that by imposing a truncated T-duality
transformation on the circular reduction of the bosonic couplings in the
heterotic theory at four- and six-derivative orders, we can calculate these
couplings in a particular YM gauge where the YM potential vanishes but its
field strength remains non-zero. Importantly, the coupling constants are
independent of the gauge choice, so these results are valid across different YM
gauge choices.
  In this work, we explore the cosmological reduction of these couplings when
the YM gauge fields belong to the Cartan subalgebra of $SO(32)$ or $E_8 \times
E_8$. We demonstrate that after applying appropriate one-dimensional field
redefinitions and total derivative terms, the couplings can be expressed in a
proposed $O(9,25)$-invariant canonical form, which is the extension of the
canonical $O(9,9)$-invariant form for just the NS-NS fields proposed by Hohm
and Zwiebach. This $O(9,25)$-invariant expression is in terms of the trace of
the first time derivative of the generalized metric, which encompasses both the
YM field and the NS-NS fields.
