The canonical Einstein metric on $G_2$ is dynamically unstable under the Ricci flow

In this note we show that the bi-invariant Einstein metric on the compact Lie group $G_2$ is dynamically unstable as a fixed point of the Ricci flow. This completes the stability analysis for the bi-invariant metrics on the compact, connected, simple Lie groups. Interestingly, $G_2$ is the only unstable exceptional group.