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



Octonionic reflection groups
Yes, such groups are interesting, see this paper by Daniel Alcock, "Reflection groups on the octave hyperbolic plane," <http://www.ma.utexas.edu/users/allcock/research/oh2.pdf>


If you write to Daniel, he will probably give you more references. 


