HIGHER ORDER JORDAN BERNSTEIN ALGEBRAS D.A.Towers Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England 1 Introduction This paper is a continuation of the study of power associative higher order Bernstein algebras initiated in [3]. We first introduce the notion of a reduced algebra and show that if A is reduced then it is a Jordan algebra. We then prove some important identities that hold in power associative higher order Bernstein algebras and deduce that such algebras satisfy the identity x 2 k+1 = ω(x)x 2 k+1 -1 for all x ∈ A. If x 1 ,...,x n belong to the algebra A we shall denote by ((x 1 ,...,x n )) the subspace spanned by x 1 ,...,x n . The symbol ⊕ will denote an algebra direct sum, whereas ˙ + will indicate a direct sum of the vector space structure alone. All algebras will be finite dimensional over a field K.