  We first study the Lipschitz spaces $\Lambda_{d}^\beta$ associated with the
Dunkl metric, $\beta\in(0,1)$, and prove that it is a proper subspace of the
classical Lipschitz spaces $\Lambda^\beta$ on $\mathbb R^N$, as the Dunkl
metric and the Euclidean metric are non-equivalent. Next, we further show that
the Lipschitz spaces $\Lambda^\beta$ connects to the Triebel--Lizorkin spaces
$\dot{ F}^{\alpha,q}_{p,{\rm D}}$ associated with the Dunkl Laplacian
$\triangle_{\rm D}$ in $\mathbb R^ N $ and to the commutators of the Dunkl
Riesz transform and the fractional Dunkl Laplacian $\triangle_{\rm
D}^{-\alpha/2}$, $0<\alpha<\textbf{N}$ (the homogeneous dimension for Dunkl
measure), which is represented via the functional calculus of the Dunkl heat
semigroup $e^{-t\triangle_{\rm D}}$. The key steps in this paper are a finer
decomposition of the underlying space via Dunkl metric and Euclidean metric to
bypass the use of Fourier analysis, and a discrete weak-type Calder\'on
reproducing formula in these new Triebel--Lizorkin spaces $\dot{
F}^{\alpha,q}_{p,{\rm D}}$.
