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Kim proved in 2000 that the octonary diagonal form \\[\n  f=x_1^2+\\cdots+x_4^2+\\varepsilon_+(x_5^2+\\cdots+x_8^2) \\] is universal over $\\mathcal{O}_K$ whenever $D=n^2-1$ is squarefree. We complete Kim's result to an if-and-only-if classification: $f$ is universal if and only if $D=n^2-1$ for some $n\\ge2$, or $D=n^2-4$ for some odd $n\\ge3$, in both cases subject to squarefreeness. 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B. M. Kim proved in 2000 that the octonary diagonal form \\[\n  f=x_1^2+\\cdots+x_4^2+\\varepsilon_+(x_5^2+\\cdots+x_8^2) \\] is universal over $\\mathcal{O}_K$ whenever $D=n^2-1$ is squarefree. We complete Kim's result to an if-and-only-if classification: $f$ is universal if and only if $D=n^2-1$ for some $n\\ge2$, or $D=n^2-4$ for some odd $n\\ge3$, in both cases subject to squarefreeness. 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