  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 $\mathrm{SO}(3,\mathbb{C})$-orbit
decomposition of the flag variety of $\mathrm{SL}_3(\mathbb{C})$.
