L8a. The group of generalized diffeomorphisms.

Generalized diffeomorphisms are orthogonal automorphisms $F$ of $T+T^*$ covering a diffeomorphism $f$ and preserving the Courant bracket.

We have proved that the group of generalized diffeomorphisms is the semidirect product
$$\textrm{Diff}(M) \subset \Omega^2_{cl}(M),$$ where the diffeomorphisms $f$ act by pushforward on $T + T^*$, and a closed $2$-form $B$ acts as a $B$-field: $X+\xi\mapsto X+\xi+i_X B$. The $B$-fields are a new symmetry, not present in classical geometry, and they will play a very important role.

Note that a diffeomorphism and a $B$-field do not commute: $$\exp(B) \circ g_*= g_* \exp(g^*B) .$$

If you look at automorphisms  of  $T$ covering a diffeomorphism $f$ and preserving the Lie bracket, you only get the differential of the diffeomorphisms, so this approach is consistent with the symmetries of classical differential geometry.