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Is there a notion of octonionic structure on a Lie group? In the same way as there is one for complex and quaternionic

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    $\begingroup$ The most sensible analogue, at least until someone mentions a better one, would be a $G_2$-structure, since $G_2 = Aut(\mathbb{O})$, as a normed division algebra. $\endgroup$
    – David Roberts ♦
    Nov 1, 2016 at 23:08
  • $\begingroup$ What's called "octonionic plane" is rather related to $F_4$, isn't it? $\endgroup$
    – YCor
    Nov 2, 2016 at 0:18
  • $\begingroup$ $\mathbb{OP}^2 = F_4/Spin(9)$, see eg math.ucr.edu/home/baez/octonions/node15.html#F4 $\endgroup$
    – David Roberts ♦
    Nov 2, 2016 at 0:31
  • $\begingroup$ The octonionic plane has isometry group $F_4$, but collineation group $E_6$. I am not sure that these facts are relevant to the question. $\endgroup$
    – Ben McKay
    Nov 2, 2016 at 8:41

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