Central Weyl involutions on Fano–Mukai fourfolds of genus 10

It is known that every Fano–Mukai fourfold X of genus 10 acted upon by an involution $$\tau $$ which comes from the center Weyl group simple algebraic type $$\textrm{G}_2$$ , see Prokhorov and Zaidenberg (Eur J Math 4(3):1197–1263 2018, Eur 8:561–572, 2022). This uniquely defined up to conjugation in $${\text {{Aut}}}(X)$$ . In this note we describe set fixed points surface scroll swept out -invariant lines.