Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V  . Let XX denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X   with respect to the basis (ii) is tridiagonal. We show that XX is spanned byI,A,A∗,AA∗,A∗Aand these elements form a basis for XX provided the dimension of V is at least 3.