I'm far to be an expert in representation theory and in GAP, but I'm
looking for a Chevalley module V for G=SO(4,C), taht is to say a faithful
finite dimensional SO(4,C)-module such that:
1. V contains no one-dimensional G-modules
2. any proper connected closed subgroup H $\in$ G leaves a one-dimensional
subspace W $\in$ V invariant.
I'd be interested in an injection of the representation of so(4,C) by the
set of matrices X such that tX.M+M.X=0, with M=[[0,I_2],[I_2,0]], in gl(V)
because I'm looking actually for the image of a regular pair of generators
of so(4,C) [which I know for the previous representation] in a Chevalley
module for SO(4,C).
Hoping it may interest some of you too,