Given a semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$ (and all the other necessary data), I want to be able to write down a matrix representation for $V(\lambda)$, the unique irreducible rep of $\mathfrak{g}$ of weight $\lambda$.
Here's my work for $\mathfrak{g}=\mathfrak{sl}_3(k)$ ($k=\overline{k}$, char $k=0$), with CSA diagonal matrices, with Borel the upper triangular matrices, and dominant integral weight $\lambda=2\lambda_1$, where $(\lambda_1=L_1,\lambda_2=L_1+L_2=-L_3)$ are fundamental dominant weights for the base ($\alpha=L_1-L_2$, $\beta=L_2-L_3$). (Here $L_i$ is the linear functional on diagonal matrices of $\mathfrak{gl}_n$ dual to $e_{ii}$).
This rep has 6 nontrivial weights which each have a one-dimensional weight space. They are $2\lambda_1,-\lambda_1,\lambda_2,-2\lambda_2,\lambda_1-\lambda_2$, and $2\lambda_2-2\lambda_1$.
To get matrices, we need a basis. So let $v$ be the maximal vector, and let $y_1=e_{21}$, $y_2=e_{32}$. Then it's not hard to check get that a basis for $V$ is given by $(v,y_1v,y_2y_1v,y_1^2v,y_2y_1^2v,y_2^2y_1^2v)$ (with this ordering).
Now, let's try to compute the matrix for $y_1$. It sends the first basis vector to the second basis vector, the second basis vector to the fourth basis vector, and the third basis vector to $y_1y_2y_1v=y_2y_1^2v+[y_1,y_2]y_1v$ which is in the weight space $V_{-\lambda_1}$, so it should be some constant multiple of $y_2y_1^2v$. My question is, $\textbf{how do we find this constant?}$ Or equivalently, $\textbf{how do we figure out how $[y_1,y_2]$ acts on }$ $y_1v$?
Your help is deeply appreciated!