  The corrections to the tree-level effective action for the bosonic string up
to second order in $\alpha'$ are fixed by requiring its dimensional reduction
to $26-d$ dimensions to be compatible with $O(d,d)$ symmetry. The result is in
agreement with the literature, but takes a simpler form than previously know
expressions. We identify some structures in the Lagrangian which appear at
least at the first three orders in $\alpha'$.
