[GAP Forum] subalgebras of simple lie algebras

[mim_ at op.pl]

Hello R.N.,

I have not yet obtained nice matrix generators of compact e8. I am working on it. I have them for e7 finally. Unfortunately simple lie algebras in GAP are not compact. My intuitions are following - obtained from Baez work "Octonions" and others.

Assume we have decomposition of e8 base:
e8=Tri(O)+24*8+Tri(O)
Tri(O) is so8 in notation from Barton, Sudbery (see Baez for detail).

We know that 
f4=Tri(O)+3*8

So if we have first decomposition then
e8=Tri(O)+3*8 +21*8+Tri(O)
We decompose second Tri(O) as
Tri(O)=so8=g2+2*7

2*7 is generated by left and right octonion multiplications
g2 are derivations of O.

Then we obtain decompostion of e8 into F4+G2+ 21*8+2*7.

Regards,
Marek Mitros