Is chiral dark matter present in other octonion-based models?

[arivero]

This article of Dubois-Violette

http://www.sciencedirect.com/science/article/pii/S0550321316300566
https://arxiv.org/abs/1604.01247

suggests that the diagonal in the exceptional jordan algebra H3(O) should be interpreted as chiral dark matter. Is it the same in other octonion-based models, or do they cancel the diagonal somehow?

 

[arivero]

Shane Farnsworth also insinuates the diagonal in his thesis
https://inspirehep.net/record/1419192/files/Farnsworth_Shane.pdf

it is natural to ask: ‘does F4 have a subgroup which leaves the three diagonal real elements, as well as the real element and one imaginary element from each off-diagonal octonion invariant?’ The subgroup is once again exactly SU(3). In other words, on breaking down from F4 to SU(3) one obtains three ‘quark’ triplets, three ‘lepton’ singlets, and three ‘Majorana’ like dark matter candidates.