Noncommutative Jordan algebra

In algebra, a noncommutative Jordan algebra is an algebra, usually over a field of characteristic not 2, such that the four operations of left and right multiplication by "x" and "x"2 all commute with each other. Examples include associative algebras and Jordan algebras.
Over fields of characteristic not 2, noncommutative Jordan algebras are the same as flexible Jordan-admissible algebras, where a Jordan-admissible algebra, introduced by and named after Pascual Jordan, is a (possibly non-associative) algebra that becomes a Jordan algebra under the product "a" ∘ "b" = "ab" + "ba". 