The classical theory of
nondegenerate quadratic forms permitting composition has recently been generalized
in several directions: Kunze and Scheinberg considered degenerate forms on
alternative algebras over fields of characteristic ≠2; Petersson and Racine briefly
considered nondegenerate forms over general rings of scalars; the generalized
Cayley-Dickson algebras of dimension 2n carry a scalar involution, but are not
alternative and do not admit composition for n > 3. In this paper we study general
algebras with scalar involution (where all norms xx∗ and traces x + x∗ are scalars)
over arbitrary rings of scalars. We locate these among all degree 2 algebras, and
derive conditions for them to be flexible, alternative, or composition algebras. We
consider Cayley elements and Cayley birepresentations, recovering the results of
Kunze and Scheinberg on radicals of norm forms. We also investigate the
Cayley-Dickson doubling process for constructing new scalar involutions out of old
ones.
