  This paper delves into the Terwilliger algebra associated with the ordered
Hamming scheme, which extends from the wreath product of one-class association
schemes and was initially introduced by Delsarte as a natural expansion of the
Hamming schemes. Levstein, Maldonado and Penazzi have shown that the
Terwilliger algebra of the Hamming scheme of length $n$ is the $n$-fold
symmetric tensor algebra of that of the one-class association scheme.
Furthermore, Bhattacharyya, Song and Tanaka have established that the
Terwilliger algebra of the wreath product of a one-class association scheme is
a direct sum of the ``primary'' subalgebra and commutative subalgebras. This
paper extends these findings to encompass both conclusions.
