The Hermitian level of composition algebras (Q1865603)

Let \(A\) be a nonassociative unital ring and let \(\tau\) be an involution on \(A\), i.e., an anti-automorphism of period~\(2\). The hermitian level of \((A,\tau)\), denoted \(S_{h}(A,\tau)\), is the smallest number of terms in a representation of \(-1\) as a sum of elements of the form \(\tau(a)a\) (with the convention that \(S_{h}(A,\tau)=\infty\) if no such representation exists). The case where \(A\) is a quaternion algebra was considered in \textit{D. W. Lewis} [J. Algebra 115, 466--480 (1988; Zbl 0644.10017)] and \textit{A. Serhir} [Commun. Algebra 25, 2531--2538 (1997; Zbl 0885.11029)]. In the present paper, the authors extend the results of Lewis and Serhir by investigating the case where \(A\) is an arbitrary composition algebra over a ring \(R\) where \(2\) is invertible. The main results are obtained for semilocal rings \(R\): the hermitian level of the standard involution is then shown to be a power of \(2\) when it is finite, and any power of \(2\) occurs as the hermitian level of a non-standard involution which is the identity on \(R\). Some bounds are obtained in the case where the involution does not restrict to the identity on \(R\).