\Sigma\right)\psi \ , \ \Sigma:= -i\Omega(\Theta^{-1})^{\mu\nu} [\Gamma_\mu ,\Gamma_{\nu+4}] \label{Sigma1}$$ with $\Delta = \partial^\mu \partial_\mu$. As algebra we chose the Moyal algebra $\mathbb{R}^4_\Theta$: $$\mathcal{A}_4 = \mathbb{R}_\Theta^4=(\mathcal{S}(\mathbb{R}^4),\star)$$ where $(\mathcal{S}(\mathbb{R}^4),\star)$ is the algebra of the Schwartz functions on $\mathbb{R}^4$, with the Moyal product $$(f \star g)(x) = \int \int d^4 y \frac{d^4 k}{(2\pi)^4} f(x+\frac{1}{2}\Theta \cdot k) g(x+y) e^{i\langle k,y\rangle} \ ,\ f,g \in \mathcal{A}_4 \label{star}$$ The representation of the algebra $\mathcal{A}_4$ on $\mathcal{H}_4$ is by component-wise diagonal Moyal product [@moyal-triple] $\star : \mathcal{A}_4 \times \mathcal{H}_4 \to \mathcal{H}_4$. The Moyal product can be extended to constant functions using another representation of the product with the integral representation of the Dirac distribution. Taking in account, for smooth spinors, the identity $2x^\mu\psi=x\star\psi+\psi\star x$ and the relation $$[x^\nu,f]_\star = i\Theta^{\nu\rho} \partial_\rho \label{comxf}$$ we compute the commutator of that action with the Dirac operator $$\begin{array}{l}
\mathcal{D}_4(f\star\psi)-f\star(\mathcal{D}_4 \psi) \\
= i\Gamma^\mu ((\partial_\mu f) \star\psi + f\star \partial_\mu \psi) + \frac{1}{2} \Omega\Gamma^{\mu+4} (\tilde{x}_\mu\star(f \star \psi) + (f \star \psi) \star \tilde{x}_\mu) \\
-i\Gamma^\mu f \star \partial_\mu \psi - \frac{1}{2} \Omega\Gamma^{\mu+4}(f\star( \tilde{x}_\mu\star\psi)+f\star(\psi \star \tilde{x}_\mu))\\
= i(\Gamma^\mu + \Omega\Gamma^{\mu+4} )(\partial_\mu f ) \star \psi .
\end{array} \label{d4-com}$$ The previous commutator confirms that $(\mathcal{A}_4,\mathcal{H}_4,\mathcal{D}_4)$ satisfy the main[^2] axioms of spectral triple, in fact 