Involutions on composition algebras

AbstractInvolutions on composition algebras over rings where 2 is invertible are investigated. It is proved that there is a one-one correspondence between non-standard involutions of the first kind, and composition subalgebras of half rank. Every non-standard involution of the first kind is isomorphic to the natural generalization of Lewis's hat-involution [L]. Any involution of the second kind on a composition algebra C over a quadratic etale R-algebra S can be written as the tensor product of the standard involution of a unique R-composition subalgebra of C and the standard involution of S/R. The latter generalizes a well-known theorem of Albert on quaternion algebras with unitary involutions