  It is known that every Fano-Mukai fourfold X of genus 10 is acted upon by an
involution $\tau$ which comes from the center of the Weyl group of the simple
algebraic group of type ${\rm G}_2$. This involution is uniquely defined up to
conjugation in the group Aut(X). In this note we describe the set of fixed
points of $\tau$ and the surface scroll swept out by the $\tau$-invariant
lines.
