Representation matrices for transformed structure constants

Question

Let $M_i$ with \begin{equation} [M_i,M_j] = \sum_{k=1}^N f_{i,j,k} \, M_k \qquad i,j=1,\ldots,N \end{equation}
denote a $d$-dimensional representation of a Lie algebra $\cal{L}$ with structure constants $f_{i,j,k}$. The structure constants are not unique and $f_{i,j,k}$ may be replaced by \begin{equation} \tilde{f}_{i,j,k}\equiv\sum_{n,m,s=1}^N A_{i,n}\,A_{j,m}\,(A^{-1})_{s,k}\,f_{n,m,s} \end{equation} with an arbitrary, invertible $N{\times}N$ matrix $A$. I'm looking for a method to construct from $M_i$ the $d$-dimensional representation matrices $\tilde{M}_i$ such that \begin{equation} [\tilde{M}_i,\tilde{M}_j] = \sum_{k=1}^N \tilde{f}_{i,j,k} \, \tilde{M}_k \qquad i,j=1,\ldots,N \end{equation}

Answer 1

Consider $\tilde{M}_m\equiv \sum_i A_{m,i} M_i$, and further $$ [\tilde{M}_m,\tilde{M}_n]= \sum_{i,j} A_{m,i} A_{n,j} [M_i,M_j]\\ = \sum_{i,j,k} A_{m,i} A_{n,j} f_{i,j,k} M_k = \sum_{i,j,k,s,l} A_{m,i} A_{n,j} f_{i,j,k} A^{-1}_{k,s} A_{s,l} M_l \\ = \sum_{i,j,k,s} A_{m,i} A_{n,j} f_{i,j,k} A^{-1}_{k,s} \tilde M_s \\ = \sum_{s} \tilde f_{m,n,s} \tilde M_s ~~ . $$