RAM Cholesky Decomposition

I am playing around with different kinds of twin models which I try to implement using the RAM-matrix approach, but not defining the expectation object with mxExpectationRAM(), but with mxExpectationNormal(). My idea is to create the A, S (one for MZs and one for DZs) and F matrices as well as the Identity matrix with the mxMatrix() functions and then creating the expected covariance matrices for MZs and DZs with the mxAlgebra() function following the RAM matrix formula.

It may be some more coding compared to the "normal" matrix approach, but my hope is that the model specified this way can be easily extended: You can simply add covariates into the covariance matrix, define the ACE decomposed variable as a latent factor etc.. I am aware that you can do the same things with the "normal" matrix approach, but following the RAM logic you just have to modify the three matrices accordingly and the formula to create the expected covariance matrices remains the same.

For a first univariate model this works fine. My questions is about implementing a bivariate model. Leohlin (1996; doi: 10.1007/BF02361160) presents different parametrizations of the same multivariate model. I think the correlated factor solution where you specify the covariances between the ACE factors instead of the cross paths can be implemented without problems with the RAM-matrices. But I am not sure how to implement the Cholesky parametrization using the RAM matrices. I am aware of the problems of the Cholesky approach as discussed in Verhulst et al. (2019; doi: 10.1007/s10519-018-9942-y), but for the sake of completeness I am interested in implementing the Cholesky approach with the RAM matrices.

EDIT: To make my point more clearly: Just filling the A matrix with the path coefficients is straightforward. But I am not sure if it is right because there is no explicit multiplication of two lower triangluar matrices. So I was wondering whether I have to rebuild the A matrix in order to ensure this multiplication. But maybe I am on the wrong track here?