It is known that every Fano-Mukai fourfold X of genus 10 is acted upon by an involution τ which comes from the center of the Weyl group of the simple algebraic group of type G 2 , see [PZ18, PZ22]. This involution is uniquely defined up to conjugation in the group Aut(X). In this note we describe the set of fixed points of τ and the surface scroll swept out by the τ-invariant lines. Contents 14 3.3. The case Aut 0 (V) = G a × G m 15 4. Smoothness of Π 16 5. Linear nondegeneracy of Π in P 12 18 5.1. The first observations 18 5.2. Linear nondegeneracy of Π 19 6. Normal bundles of the fixed curves 21 References 23