  We show that any continuous $\mathbf{C}$-linear Lie algebra splitting of the
symbol map from the Atiyah algebra of a vector bundle on a complex manifold is
given by a differential operator of order at most the rank of the bundle plus
one. Bundles equipped with such a splitting can be thought of as
\emph{infinitesimally equivariant} bundles, and our theorem implies these are,
in a certain sense, in a categorical formal neighbourhood of vector bundles
with a flat connection.
