# Octonionic reflection groups

Consider Leech lattice definition provided by Wilson (Octonions and Leech lattice, 2008). There are 819 E8 sublattices defined by

$(2\lambda, 0, 0);$ $(\lambda \overline{s}, (\lambda \overline{s}) j, 0);$ $( (\lambda s)j, \lambda k, (\lambda j) k )$

where $\lambda$ span 240 vectors of E8 lattice, j,k are 16 base octonions (plus, minus), and s is -1+sum of imaginary unit octonions. (I am testing LaTeX here) See page 3, chapter 3 of Wilson paper. I wonder what is the subgroup of $Co_0$ generated by 819 reflections in 8-dim planes spanned by those E8 sublattices. They could be considered as octonion reflections. And as such they are elements of F4 Lie group being automorphism of $OP^2$.

My questions is following. Has anyone tried to extend definition of complex reflection and quaternion reflection to octonion reflection. In such definition Conway group $Co_0$ would be octonion reflection group i.e. it is generated by reflections in 8-dim planes in 24-dim Euclidean space.

In general when order 2 element in abstract group - called involution - can be considered as reflection ? I know involution is algebraic notion while reflection is geometric. But geometry is something which make group theory interesting.

Regards, Marek

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