  We classify states of four rebits, that is, we classify the orbits of the
group $\widehat{G}(\mathbb R) = \mathrm{\mathop{SL}}(2,\mathbb R)^4$ in the
space $(\mathbb R^2)^{\otimes 4}$. This is the real analogon of the well-known
SLOCC operations in quantum information theory. By constructing the
$\widehat{G}(\mathbb R)$-module $(\mathbb R^2)^{\otimes 4}$ via a $\mathbb
Z/2\mathbb Z$-grading of the simple split real Lie algebra of type $D_4$, the
orbits are divided into three groups: semisimple, nilpotent and mixed. The
nilpotent orbits have been classified in Dietrich et al. (2017), yielding
applications in theoretical physics (extremal black holes in the STU model of
$\mathcal{N}=2, D=4$ supergravity, see Ruggeri and Trigiante (2017)). Here we
focus on the semisimple and mixed orbits which we classify with recently
developed methods based on Galois cohomology, see Borovoi et al. (2021). These
orbits are relevant to the classification of non-extremal (or extremal
over-rotating) and two-center extremal black hole solutions in the STU model.
