gr.group theory - Leech lattice decomposition

Hi again,

Now I think I could perform decomposition using Wilson definition of Leech lattice using octonions (2008). I this definition Leech lattice is easily seen as union of 819 E8 sublattices;
819 = 3*(1+16+16*16).
Having this we can decompose each E8 lattice into crosses. Last step (little vague) would be to find decomposition of 819 E8 lattices into 273 triples.

But now I am struggling with following problem. Consider 2A class in Co1 having 819*759*75 elements. Each element a from 2A have two representatives in Co0. Element a corresponds to E8 sublattice in Leech defined as {v: av=-v} where I call by a also proper preimage in Co0. Now the opposite having E8 sublattice L in Leech I want to find element a(E8) in 2A class. My straightforward function build c*d*c^-1 where d is diagonal matrix changing sign in octad [1..8]. But I have not obtained Co0 element.

My goal is to find relation between Order(ab) for a,b in 2A and corresponding geometry of two E8 sublattices. The Order(ab) can be 2,3,4,5,6.

Take any other sporadic group g and certain conjugacy class cg of order 2 elements. Is it known possible values of Order(ab) for a,b in cg ?

Regards,
Marek