The question is, does the fact $ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)^{2}=0, \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 1 & 0 &0 \end{array}\right)^{2}=0$ influence $SU(3)$'s irreducible representation?
Background: To simplify I am working over the case of $SU(3)$. For all $sl_{n}\mathbb{C}$ any irreducible representation is generated by repeated applying negative weight elements to the highest weight vector $v$ (standard reference, Serre). But when I try to draw the weight diagram starting from any given highest weight vector $v$ for $H$, say (10, 20), then repeating applying the negative weight element like this one $ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)$ always give me zero because $ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)^{2}=0$. Now elementary analysis showed that $v$'s image under the action of $ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)$ and $ \left(\begin{array}{ccc} 0 & 0 & 0\\ 1 & 0& 0\\ 0 & 0 &0 \end{array}\right)$ actually generates $V$. So $V$'s weight diagram must be very small, which is counter-intuitive for me. I am hoping I may claim there exists irreducible subrepresentation of $sl_{2}\mathbb{C}$ by similar arguments using either of the matrices repeatedly until the image vanishes. I think I must be confused with something very basic, like the embedding of $sl_{3}\mathbb{C}$ in $gl_{n}\mathbb{C}$ or something.