  The Bianchi identities for bosonic fluxes in supergravity can receive higher
derivative quantum and string corrections, the most well known being that of
Heterotic theory $d H = \tfrac{1}{4}\alpha'(\text{tr } F^2 - \text{tr } R^2)$.
Less studied are the modifications at order $R^4$ that may arise, for example,
in the Bianchi identity for the seven-form flux of M theory compactifications.
We argue that such corrections appear to be incompatible with the exceptional
generalised geometry description of the lower order supergravity, and seem to
imply a gauge algebra for the bosonic potentials that cannot be written in
terms of an (exceptional) Courant bracket. However, we show that this algebra
retains the form of an $L_{\infty}$ gauge field theory, which terminates at a
level ten multibracket for the case involving just the seven-form flux.
