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 x2 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.