In this paper algebras with identity, over ordered fields, which possess a (weak) submultiplicative positive bilinear form are studied. They are called (weak) subdecomposition algebras. It is proved that every weak subdecomposition algebra is a simple quadratic algebra with no nontrivial nilpotents or idempotents. If a weak subdecomposition algebra is also flexible, then it is a noncommutative Jordan algebra and an algebraic involution can be defined in a natural way. Every subdecomposition algebra is automatically flexible. There exist simple examples of weak subdecomposition algebras which are not flexible. Alternative weak subdecomposition algebras can be completely classified. Some structure theorems for weak subdecomposition algebras with large nucleus are also given.