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The Greenberg functor $\\mathcal{F}$ associates to $\\mathbf{G}$ a linear algebraic group $G:=(\\mathcal{F}\\mathbf{G})(k)$ over $k$, such that $G\\cong\\mathbf{G}(A)$. We prove that if $\\mathbf{G}$ is a reductive group scheme over $A$, and $\\mathbf{T}$ is a maximal torus of $\\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. 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