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In this paper, we give $\mathbb{Z}\left[1/2\right]$-forms of$\mathrm{SO}(3,\mathbb{C})$-orbits in the flag variety of$\mathrm{SL}_3(\mathbb{C})$. We also prove that they give a$\mathbb{Z}\left[1/2\right]$-form of the $\mathrm{SO}(3,\mathbb{C})$-orbitdecomposition of the flag variety of $\mathrm{SL}_3(\mathbb{C})$.

$\mathrm{SO}(3)$-homogeneous decomposition of the flag scheme of $\mathrm{SL}_3$ over $\mathbb{Z}\left[1/2\right]$