Bruhat order in the Toda system on $\mathfrak{so}(2,4)$: an example of non-split real form

In our previous papers we described the structure of trajectories of the symmetric Toda system on normal real forms of various Lie algebras and showed that it was totally determined by the Hasse diagram of the Bruhat order on the corresponding Weil group. This note deals with the simplest non-split real Lie algebra, $\mathfrak{so}(2,4)$. It turns out, that the phase diagram in this case is also closely related with the Bruhat order of the relative Weyl group, but a bit more information is necessary to describe the dimensions of the trajectory spaces.