Composition algebras satisfying certain identities

Even though the class of power associative algebras is huge, only Hurwitz algebras appear in the case of composition algebras, not necessarily unital, over fields of characteristic different from two. Under this assumption and other conditions weaker than power associativity, we study in this note composition algebras of arbitrary dimension. First, a characterization of Hurwitz algebras is given when the ground field has at least five elements. More cases are considered. So, by using conditions on powers of every element, we characterize when a composition algebra over a field of characteristic different from 2 and 3 and with at least 7 elements is Hurwitz or its norm is associative. Similar characterizations are given for composition algebras that are Hurwitz or standard II (respectively Hurwitz or standard III).