Let $\mathfrak{g}$ be the complex Lie algebra $\mathfrak{sl}_3(\Bbb{C})$. Consider adjoint representation $\textrm{ad} : \mathfrak{sl}_3(\Bbb{C}) \to \textrm{gl}(\mathfrak{g})$. $\mathfrak{g}$ has the usual complex basis
$\begin{eqnarray*} H_1 &=& \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{array}\right), H_2 &=& \left(\begin{array}{ccc} 0& 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right),\\ X_1 &=& \left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right), X_2 &=& \left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right), X_3 &=& \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right),\\ Y_1&=& \left(\begin{array}{ccc} 0& 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right), Y_2 &=& \left(\begin{array}{ccc} 0& 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right), Y_3 &=& \left(\begin{array}{ccc} 0& 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)\end{eqnarray*}.$
Now by restricting $\textrm{ad}$ to just the vectors $H_1,X_1$ and $Y_1$ I can get an 8 dimensional representation of $\mathfrak{sl}_2(\Bbb{C})$. Suppose I wish to decompose this representation into irreducibles. Now I have checked that $\textrm{span}\{X_2,X_3\}$ and $\textrm{span}\{Y_2,Y_3\}$ are irreducible 2 - dimensional subrepresentations. Now there are still the vectors $H_1,H_2,X_1,Y_1$ whose span I have tried to check is irreducible. If we write down
$\textrm{ad}_{H_1}, \textrm{ad}_{X_1}, \textrm{ad}_{Y_1}, $
in the basis $H_1,H_2,X_1,Y_1,X_2,X_3,Y_2,Y_3$ (in this order) we get
$\begin{eqnarray*} \textrm{ad}_{H_1} &=& \left(\begin{array}{cccc|cc|cc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0& 0 & 2 & 0 \\ 0& 0& 0 & -2 \\ \hline &&&& -1 & 0 \\ &&&& 0 & 1 \\ \hline &&&&&&& 1 & 0 \\&&&&&&& 0 & -1 \end{array}\right) \textrm{ad}_{X_1} &=& \left(\begin{array}{cccc|cc|cc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ -2& 1 & 0 & 0 \\ 0& 0& 0 & -2 \\ \hline &&&& 0 & 0 \\ &&&& 1 & 0 \\ \hline &&&&&&& 0 & -1 \\&&&&&&& 0 & 0 \end{array}\right) \\ \textrm{ad}_{Y_1} &=& \left(\begin{array}{cccc|cc|cc} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0& 0 & 0 & 0 \\ 2& -1& 0 & 0 \\ \hline &&&& 0 & 1 \\ &&&& 0 & 0 \\ \hline &&&&&&& 0 & 0 \\&&&&&&& 1 & 0 \end{array}\right). \\ \end{eqnarray*}$
From looking at the first $4 \times 4$ block in each matrix it seems that the span $\{H_1,H_2,X_1,Y_1\}$ is irreducible. How do I prove this formally in an elegant way without bashing?