what would be the best model to test GxE in a multivariate model?

Hi,

I would appreciate some advice on what model I need to use in order to test some specific hypotheses. I have suggested some ways to do this, but not sure if it is doable or correct

I have three traits, with phenotypic correlations of around .40.

Hypotheses:
1- large genetic correlation between trait 1 and trait 2 and 3
2- Much of the genetic variance component of trait 2 and trait 3 can be explained by the interaction between environmental variable B and common genetic factors of trait 1&2&3.

I thought I could maybe try the following analyses:

1-There is a genetic correlation between these 3 factors (Cholesky correlated factors solution)

2a- fit and compare 3 models: Cholesky decomposition with trait 1 ordered as first variable, A contributing to trait 2 and 3; common pathway model; independent pathway model)

2b- Assuming the common pathway model is the best fitting model from 2a: run a common pathways x E model, whereby genetic variance components of trait 2 and 3 are replaced by trait1 A xE interaction?

Does this make any sense, or am I getting too confused?

Thank you!