Corollary

Let A be a ring endowed with an antiautomorphism J, E a left A-module, Φ a sesquilinear form for J on E × E whose associated mappings are bijective, and u an automorphism of the A-module E. In order that u leave Φ invariant (that is to say that one have Φ(u(x), u(y)) = Φ(x, y) whatever x, y in E may be), it is necessary and sufficient that the two adjoints of u be equal and that one have u^* = u^{-1}.

This follows at once from prop. 8.