Dual connection to flat connection is flat

Statement

Statement in the language of connections

Suppose $M$ is a differential manifold, $E$ is a vector bundle over $M$, and $\nabla$ is a connection on $M$. Suppose $E^*$ is the dual bundle to $E$ and $\nabla^*$ is the dual connection to $\nabla$.

Then, if $\nabla$ is a flat connection (i.e., the Riemann curvature tensor $R_\nabla$ is identically zero), $\nabla^*$ is also a flat connection.

Statement in the language of modules over the connection algebra

Fill this in later

Facts used

1. Formula for curvature of dual connection