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Building upon previous works, it is shown that two minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(6)$ and two minimal right ideals of $\mathbb{C}\ell(4)$ transform as one generation of leptons and quarks under the gauge symmetry $SU(3)_C\times U(1)_{EM}$ and $SU(2)_L\times U(1)_Y$ respectively. The $SU(2)_L$ weak symmetries are naturally chiral. Combining the $\mathbb{C}\ell(6)$ and $\mathbb{C}\ell(4)$ ideals, all the gauge symmetries of the Standard Model, together with its lepton and quark content for a single generation are represented, with the dimensions of the minimal ideals dictating the number of distinct physical states. The combined ideals can be written as minimal left ideals of $\mathbb{C}\ell(6)\otimes\mathbb{C}\ell(4)\cong \mathbb{C}\ell(10)$ in a way that preserves individually the $\mathbb{C}\ell(6)$ structure and $\mathbb{C}\ell(4)$ structure of physical states. This resulting model includes many of the attractive features of the Georgi and Glashow $SU(5)$ grand unified theory without introducing proton decay or other unobserved processes. Such processes are naturally excluded because they do not individually preserve the $\mathbb{C}\ell(6)$ and $\mathbb{C}\ell(4)$ minimal ideals.