Let $D\geq 1$ and $q\geq 3$ be two integers. Let $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. Let ${\mathcal T}(D)$ denote the Terwilliger algebra of $H(D)$. Let $V(D)$ denote the standard ${\mathcal T}(D)$-module. Let $ω$ denote a complex scalar. We consider a unital associative algebra $\mathfrak K_ω$ defined by generators and relations. The generators are $A$ and $B$. The relations are $A^2 B-2 ABA +B A^2 =B+ωA$, $B^2A-2 BAB+AB^2=A+ωB$. The algebra $\mathfrak K_ω$ is the case of the Askey-Wilson algebras corresponding to the Krawtchouk polynomials. The algebra $\mathfrak K_ω$ is isomorphic to ${\rm U}(\mathfrak{sl}_2)$ when $ω^2\not=1$. We view $V(D)$ as a $\mathfrak{K}_{1-\frac{2}{q}}$-module. We apply the Clebsch-Gordan rule for ${\rm U}(\mathfrak{sl}_2)$ to decompose $V(D)$ into a direct sum of irreducible ${\mathcal T}(D)$-modules.