Commutativity of products for adjoint operators

Let L deonote a finite dimensional semisimple Lie algebra over an arbitrary field of characteristic zero, and U = U(L) its universal enveloping algebra. Using the structure constants of L, one can define a vector product in the space Hom/sub L/(L,U) which makes it an associative algebra with identity. We prove here an earlier conjecture of Okubo and Myung that this algebra is commutative. We also define a scalar product which maps two elements of Hom/sub L/(L,U) into the center of U and show it to be commutative