Geodesic
Octonionic Cayley spinors and $E_6$
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 193-207

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Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group $E_6$, and of its subgroups. We are therefore led to a description of $E_6$ in terms of $3\times 3$ octonionic matrices, generalizing previous results in the $2\times 2$ case. Our treatment naturally includes a description of several important subgroups of $E_6$, notably $G_2$, $F_4$, and (the double cover of) $SO(9,1)$. An interpretation of the actions of these groups on the squares of 3-component Cayley spinors is suggested.
Classification : 17A35, 17C90, 22E70
Keywords: octonions; $E_6$; exceptional Lie groups; Dirac equation 
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Dray, Tevian; Manogue, Corinne A. Octonionic Cayley spinors and $E_6$. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 193-207. http://geodesic.mathdoc.fr/item/CMUC_2010__51_2_a4/